%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP106^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:36 AM UTC 2026
% Result : Theorem 3.07s 2.00s
% Output : Refutation 3.07s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 29 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 103 ( 31 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 730 ( 16 ~; 10 |; 0 &; 698 @)
% ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 10 con; 0-3 aty)
% Number of variables : 28 ( 0 ^; 28 !; 0 ?; 28 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
partia1993116613t_unit: $tType ).
thf(type_def_6,type,
set_Product_prod_a_a: $tType ).
thf(type_def_8,type,
partia96731725t_unit: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
partia1833973666xt_a_b: $tType ).
thf(type_def_11,type,
a: $tType ).
thf(type_def_12,type,
int: $tType ).
thf(type_def_13,type,
nat: $tType ).
thf(type_def_14,type,
product_prod_a_a: $tType ).
thf(type_def_15,type,
b: $tType ).
thf(func_def_0,type,
partia1993116613t_unit: $tType ).
thf(func_def_1,type,
partia1833973666xt_a_b: $tType ).
thf(func_def_2,type,
partia96731725t_unit: $tType ).
thf(func_def_3,type,
set_Product_prod_a_a: $tType ).
thf(func_def_4,type,
product_prod_a_a: $tType ).
thf(func_def_5,type,
set_a: $tType ).
thf(func_def_6,type,
nat: $tType ).
thf(func_def_7,type,
int: $tType ).
thf(func_def_8,type,
b: $tType ).
thf(func_def_9,type,
a: $tType ).
thf(func_def_10,type,
partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
thf(func_def_11,type,
partia1955795460t_unit: partia96731725t_unit > set_a ).
thf(func_def_12,type,
partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
thf(func_def_13,type,
units_a_Product_unit: partia96731725t_unit > set_a ).
thf(func_def_14,type,
units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
thf(func_def_15,type,
comm_g1684316527t_unit: partia96731725t_unit > $o ).
thf(func_def_16,type,
comm_g791708116xt_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_17,type,
group_a_Product_unit: partia96731725t_unit > $o ).
thf(func_def_18,type,
group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_19,type,
mult_a_Product_unit: partia96731725t_unit > a > a > a ).
thf(func_def_20,type,
mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_21,type,
one_a_Product_unit: partia96731725t_unit > a ).
thf(func_def_22,type,
one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
thf(func_def_23,type,
units_873712258t_unit: partia96731725t_unit > partia96731725t_unit ).
thf(func_def_24,type,
units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
thf(func_def_25,type,
minus_minus_int: int > int > int ).
thf(func_def_26,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_27,type,
plus_plus_int: int > int > int ).
thf(func_def_28,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_29,type,
uminus_uminus_int: int > int ).
thf(func_def_30,type,
suc: nat > nat ).
thf(func_def_31,type,
semiri2019852685at_int: nat > int ).
thf(func_def_32,type,
semiri1382578993at_nat: nat > nat ).
thf(func_def_33,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_34,type,
a_inv_a_b: partia1833973666xt_a_b > a > a ).
thf(func_def_35,type,
a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_36,type,
abelian_group_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_37,type,
add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
thf(func_def_38,type,
add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
thf(func_def_39,type,
field_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_40,type,
add_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_41,type,
more_a_b: partia1833973666xt_a_b > b ).
thf(func_def_42,type,
zero_a_b: partia1833973666xt_a_b > a ).
thf(func_def_43,type,
semiring_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_44,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_45,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_46,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_47,type,
member_a: a > set_a > $o ).
thf(func_def_48,type,
r: partia1833973666xt_a_b ).
thf(func_def_49,type,
r2: a ).
thf(func_def_50,type,
r3: a ).
thf(func_def_51,type,
r4: a ).
thf(func_def_52,type,
rel: partia1993116613t_unit ).
thf(func_def_53,type,
s: a ).
thf(func_def_54,type,
s2: a ).
thf(func_def_55,type,
s3: a ).
thf(func_def_65,type,
sK0: ( product_prod_a_a > $o ) > a ).
thf(func_def_66,type,
sK1: ( product_prod_a_a > $o ) > a ).
thf(func_def_67,type,
sK2: product_prod_a_a > a ).
thf(func_def_68,type,
sK3: product_prod_a_a > a ).
thf(func_def_69,type,
sK4: product_prod_a_a > a ).
thf(func_def_70,type,
sK5: product_prod_a_a > a ).
thf(func_def_71,type,
sK6: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_72,type,
sK7: ( product_prod_a_a > $o ) > a ).
thf(func_def_73,type,
sK8: ( product_prod_a_a > $o ) > a ).
thf(func_def_76,type,
ph10:
!>[X0: $tType] : X0 ).
thf(f1800,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1799,f787]) ).
thf(f787,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
!= $true ),
inference(backward_demodulation,[status(thm)],[f739,f731]) ).
thf(f731,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,axiom,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_6_f9) ).
thf(f739,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f664]) ).
thf(f664,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
!= $true ),
inference(flattening,[status(thm)],[f522]) ).
thf(f522,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f521]) ).
thf(f521,plain,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ),
inference(rectify,[status(thm)],[f284]) ).
thf(f284,negated_conjecture,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ),
inference(negated_conjecture,[status(cth)],[f283]) ).
thf(f283,conjecture,
member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ),
file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',conj_0) ).
thf(f1799,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1796,f731]) ).
thf(f1796,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1791]) ).
thf(f1791,plain,
( ( $true != $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) )
= $true ) ),
inference(superposition,[status(thm)],[f1222,f736]) ).
thf(f736,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f336]) ).
thf(f336,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(fool_elimination,[status(thm)],[f335]) ).
thf(f335,plain,
member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ),
inference(rectify,[status(thm)],[f11]) ).
thf(f11,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ),
file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_10_f5) ).
thf(f1222,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X1 @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ X0 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ X1 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1217]) ).
thf(f1217,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( $true != $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X1 @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ X0 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ X1 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ) ),
inference(superposition,[status(thm)],[f715,f716]) ).
thf(f716,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f476]) ).
thf(f476,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ),
inference(fool_elimination,[status(thm)],[f475]) ).
thf(f475,plain,
member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ),
inference(rectify,[status(thm)],[f12]) ).
thf(f12,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ),
file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_11_f4) ).
thf(f715,plain,
! [X2: a,X3: a,X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X2 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X1 ) ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X2 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f684]) ).
thf(f684,plain,
! [X0: a,X1: a,X2: a,X3: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X1 ) ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X2 ) ) @ ( partia206007992t_unit @ rel ) )
= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X2 ) @ ( partia206007992t_unit @ rel ) )
!= $true ) ),
inference(flattening,[status(thm)],[f683]) ).
thf(f683,plain,
! [X0: a,X1: a,X2: a,X3: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X1 ) ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X2 ) ) @ ( partia206007992t_unit @ rel ) )
= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X2 ) @ ( partia206007992t_unit @ rel ) )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ ( partia206007992t_unit @ rel ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f474]) ).
thf(f474,plain,
! [X0: a,X1: a,X2: a,X3: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ ( partia206007992t_unit @ rel ) )
= $true )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X2 ) @ ( partia206007992t_unit @ rel ) )
= $true )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X1 ) ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X2 ) ) @ ( partia206007992t_unit @ rel ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f473]) ).
thf(f473,plain,
! [X0: a,X1: a,X2: a,X3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X2 ) @ ( partia206007992t_unit @ rel ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X1 ) ) @ ( mult_a_ring_ext_a_b @ r @ X0 @ X2 ) ) @ ( partia206007992t_unit @ rel ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X4: a,X5: a,X6: a,X3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X4 ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X5 @ X6 ) @ ( partia206007992t_unit @ rel ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X6 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ X4 @ X5 ) ) @ ( mult_a_ring_ext_a_b @ r @ X4 @ X6 ) ) @ ( partia206007992t_unit @ rel ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_4_closed__rel__add) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP106^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n006.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:26:00 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP106^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Localization problem prob_1149__8998378_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Localization/prob_1149__8998378_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.44 v9.1.0, 0.38 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 339 ( 106 unt; 56 typ; 0 def)
% 0.30/0.46 % Number of atoms : 801 ( 382 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 17 ( 2 avg)
% 0.30/0.46 % Number of connectives : 3859 ( 8 ~; 1 |; 18 &;3401 @)
% 0.30/0.46 % ( 0 <=>; 431 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 16 ( 6 avg)
% 0.30/0.46 % Number of types : 11 ( 10 usr)
% 0.30/0.46 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 47 ( 46 usr; 8 con; 0-3 aty)
% 0.30/0.46 % Number of variables : 682 ( 13 ^; 653 !; 16 ?; 682 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:40:36.834
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (10)
% 0.30/0.46 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J_J,type,
% 0.30/0.46 partia1993116613t_unit: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
% 0.30/0.46 partia1833973666xt_a_b: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
% 0.30/0.46 partia96731725t_unit: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.46 set_Product_prod_a_a: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.46 product_prod_a_a: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.30/0.46 set_a: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Int__Oint,type,
% 0.30/0.46 int: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_tf__b,type,
% 0.30/0.46 b: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_tf__a,type,
% 0.30/0.46 a: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (46)
% 0.30/0.46 thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
% 0.30/0.46 partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
% 0.30/0.46 partia1955795460t_unit: partia96731725t_unit > set_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
% 0.30/0.46 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 units_a_Product_unit: partia96731725t_unit > set_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 comm_g1684316527t_unit: partia96731725t_unit > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 comm_g791708116xt_a_b: partia1833973666xt_a_b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 group_a_Product_unit: partia96731725t_unit > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 mult_a_Product_unit: partia96731725t_unit > a > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 one_a_Product_unit: partia96731725t_unit > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ounits__of_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.46 units_873712258t_unit: partia96731725t_unit > partia96731725t_unit ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.30/0.46 units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.30/0.46 minus_minus_int: int > int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.30/0.46 minus_minus_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.30/0.46 plus_plus_int: int > int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.30/0.46 plus_plus_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.30/0.46 uminus_uminus_int: int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_OSuc,type,
% 0.30/0.46 suc: nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.30/0.46 semiri2019852685at_int: nat > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.30/0.46 semiri1382578993at_nat: nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.30/0.46 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 0.30/0.46 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 0.30/0.46 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
% 0.30/0.46 abelian_group_a_b: partia1833973666xt_a_b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 0.30/0.46 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
% 0.30/0.46 add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
% 0.30/0.46 field_a_b: partia1833973666xt_a_b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 0.30/0.46 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oring_Omore_001tf__a_001tf__b,type,
% 0.30/0.46 more_a_b: partia1833973666xt_a_b > b ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 0.30/0.46 zero_a_b: partia1833973666xt_a_b > a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 0.30/0.46 semiring_a_b: partia1833973666xt_a_b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.46 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.30/0.46 collect_a: ( a > $o ) > set_a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.46 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001tf__a,type,
% 0.30/0.46 member_a: a > set_a > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_R,type,
% 0.30/0.46 r: partia1833973666xt_a_b ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_r,type,
% 0.30/0.46 r2: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_r_H,type,
% 0.30/0.46 r3: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_r_H_H,type,
% 0.30/0.46 r4: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_rel,type,
% 0.30/0.46 rel: partia1993116613t_unit ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_s,type,
% 0.30/0.46 s: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_s_H,type,
% 0.30/0.46 s2: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_s_H_H,type,
% 0.30/0.46 s3: a ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (282)
% 0.30/0.46 thf(fact_0_right__add__eq,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( add_a_b @ r @ C @ A )
% 0.30/0.46 = ( add_a_b @ r @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_add_eq
% 0.30/0.46 thf(fact_1_assms_I3_J,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ r4 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % assms(3)
% 0.30/0.46 thf(fact_2_assms_I2_J,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % assms(2)
% 0.30/0.46 thf(fact_3_assms_I1_J,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % assms(1)
% 0.30/0.46 thf(fact_4_closed__rel__add,axiom,
% 0.30/0.46 ! [R: a,S: a,R2: a,S2: a] :
% 0.30/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.30/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 0.30/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ S2 @ R ) @ ( mult_a_ring_ext_a_b @ r @ S @ R2 ) ) @ ( mult_a_ring_ext_a_b @ r @ S @ S2 ) ) @ ( partia206007992t_unit @ rel ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % closed_rel_add
% 0.30/0.46 thf(fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) @ s3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') = s \<otimes> (s'' \<otimes> s') \<otimes> s''\<close>
% 0.30/0.46 thf(fact_6_f9,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % f9
% 0.30/0.46 thf(fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ s ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (r' \<otimes> r'') = s'' \<otimes> s \<otimes> (r' \<otimes> r'')\<close>
% 0.30/0.46 thf(fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s \<otimes> r' \<otimes> r'') = s \<otimes> s' \<otimes> s'' \<otimes> (s \<otimes> s'' \<otimes> (r' \<otimes> r''))\<close>
% 0.30/0.46 thf(fact_9_f10,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % f10
% 0.30/0.46 thf(fact_10_f5,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % f5
% 0.30/0.46 thf(fact_11_f4,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % f4
% 0.30/0.46 thf(fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') = s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s' \<otimes> r \<otimes> r'' \<oplus> s \<otimes> r' \<otimes> r'')\<close>
% 0.30/0.46 thf(fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') = s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r''))\<close>
% 0.30/0.46 thf(fact_14_f7,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 0.30/0.46 = ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % f7
% 0.30/0.46 thf(fact_15_f8,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) )
% 0.30/0.46 = ( 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 @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % f8
% 0.30/0.46 thf(fact_16_f12,axiom,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 % f12
% 0.30/0.46 thf(fact_17_l__distr,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.30/0.46 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_distr
% 0.30/0.46 thf(fact_18_r__distr,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_distr
% 0.30/0.46 thf(fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062,axiom,
% 0.30/0.46 ( ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') \<ominus> s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r'')) = \<zero>\<close>
% 0.30/0.46 thf(fact_20_prod_Oinject,axiom,
% 0.30/0.46 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 0.30/0.46 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 0.30/0.46 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.30/0.46 = ( ( X1 = Y1 )
% 0.30/0.46 & ( X2 = Y2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod.inject
% 0.30/0.46 thf(fact_21_old_Oprod_Oinject,axiom,
% 0.30/0.46 ! [A: a,B: a,A2: a,B2: a] :
% 0.30/0.46 ( ( ( product_Pair_a_a @ A @ B )
% 0.30/0.46 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.30/0.46 = ( ( A = A2 )
% 0.30/0.46 & ( B = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.inject
% 0.30/0.46 thf(fact_22_local_Osemiring__axioms,axiom,
% 0.30/0.46 semiring_a_b @ r ).
% 0.30/0.46
% 0.30/0.46 % local.semiring_axioms
% 0.30/0.46 thf(fact_23_m__assoc,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % m_assoc
% 0.30/0.46 thf(fact_24_m__comm,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % m_comm
% 0.30/0.46 thf(fact_25_m__lcomm,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % m_lcomm
% 0.30/0.46 thf(fact_26_add_Om__assoc,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.30/0.46 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.m_assoc
% 0.30/0.46 thf(fact_27_add_Om__comm,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.m_comm
% 0.30/0.46 thf(fact_28_add_Om__lcomm,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 0.30/0.46 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.m_lcomm
% 0.30/0.46 thf(fact_29_local_Oright__minus__eq,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( a_minus_a_b @ r @ C @ A )
% 0.30/0.46 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.right_minus_eq
% 0.30/0.46 thf(fact_30_local_Ominus__unique,axiom,
% 0.30/0.46 ! [Y: a,X: a,Y3: a] :
% 0.30/0.46 ( ( ( add_a_b @ r @ Y @ X )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ X @ Y3 )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.minus_unique
% 0.30/0.46 thf(fact_31_add_Or__inv__ex,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( add_a_b @ r @ X @ X3 )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.r_inv_ex
% 0.30/0.46 thf(fact_32_add_Oone__unique,axiom,
% 0.30/0.46 ! [U: a] :
% 0.30/0.46 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ U @ X3 )
% 0.30/0.46 = X3 ) )
% 0.30/0.46 => ( U
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.one_unique
% 0.30/0.46 thf(fact_33_add_Ol__inv__ex,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( add_a_b @ r @ X3 @ X )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.l_inv_ex
% 0.30/0.46 thf(fact_34_add_Oinv__comm,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ Y @ X )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_comm
% 0.30/0.46 thf(fact_35_right__inv__add,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 0.30/0.46 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_inv_add
% 0.30/0.46 thf(fact_36_four__elem__comm,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a,D: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 0.30/0.46 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % four_elem_comm
% 0.30/0.46 thf(fact_37_local_Oadd_Oright__cancel,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ Y @ X )
% 0.30/0.46 = ( add_a_b @ r @ Z @ X ) )
% 0.30/0.46 = ( Y = Z ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.add.right_cancel
% 0.30/0.46 thf(fact_38_add_Om__closed,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.m_closed
% 0.30/0.46 thf(fact_39_zero__closed,axiom,
% 0.30/0.46 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % zero_closed
% 0.30/0.46 thf(fact_40_semiring__simprules_I3_J,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_simprules(3)
% 0.30/0.46 thf(fact_41_minus__closed,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_closed
% 0.30/0.46 thf(fact_42_r__zero,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_zero
% 0.30/0.46 thf(fact_43_l__zero,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_zero
% 0.30/0.46 thf(fact_44_add_Or__cancel__one_H,axiom,
% 0.30/0.46 ! [X: a,A: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( add_a_b @ r @ A @ X ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.r_cancel_one'
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.30/0.46 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: a,P: a > $o] :
% 0.30/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_47_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a] :
% 0.30/0.46 ( ( collec645855634od_a_a
% 0.30/0.46 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_48_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_a] :
% 0.30/0.46 ( ( collect_a
% 0.30/0.46 @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_49_Collect__cong,axiom,
% 0.30/0.46 ! [P: a > $o,Q: a > $o] :
% 0.30/0.46 ( ! [X3: a] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( Q @ X3 ) )
% 0.30/0.46 => ( ( collect_a @ P )
% 0.30/0.46 = ( collect_a @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_50_Collect__cong,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.30/0.46 ( ! [X3: product_prod_a_a] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( Q @ X3 ) )
% 0.30/0.46 => ( ( collec645855634od_a_a @ P )
% 0.30/0.46 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_51_add_Or__cancel__one,axiom,
% 0.30/0.46 ! [X: a,A: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ A @ X )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.r_cancel_one
% 0.30/0.46 thf(fact_52_add_Ol__cancel__one_H,axiom,
% 0.30/0.46 ! [X: a,A: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( add_a_b @ r @ X @ A ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.l_cancel_one'
% 0.30/0.46 thf(fact_53_add_Ol__cancel__one,axiom,
% 0.30/0.46 ! [X: a,A: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ X @ A )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.l_cancel_one
% 0.30/0.46 thf(fact_54_r__null,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_null
% 0.30/0.46 thf(fact_55_l__null,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_null
% 0.30/0.46 thf(fact_56_old_Oprod_Oinducts,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.30/0.46 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.30/0.46 => ( P @ Prod ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.inducts
% 0.30/0.46 thf(fact_57_old_Oprod_Oexhaust,axiom,
% 0.30/0.46 ! [Y: product_prod_a_a] :
% 0.30/0.46 ~ ! [A4: a,B3: a] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( product_Pair_a_a @ A4 @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.exhaust
% 0.30/0.46 thf(fact_58_Pair__inject,axiom,
% 0.30/0.46 ! [A: a,B: a,A2: a,B2: a] :
% 0.30/0.46 ( ( ( product_Pair_a_a @ A @ B )
% 0.30/0.46 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.30/0.46 => ~ ( ( A = A2 )
% 0.30/0.46 => ( B != B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pair_inject
% 0.30/0.46 thf(fact_59_prod__cases,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.30/0.46 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.30/0.46 => ( P @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases
% 0.30/0.46 thf(fact_60_surj__pair,axiom,
% 0.30/0.46 ! [P2: product_prod_a_a] :
% 0.30/0.46 ? [X3: a,Y4: a] :
% 0.30/0.46 ( P2
% 0.30/0.46 = ( product_Pair_a_a @ X3 @ Y4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_pair
% 0.30/0.46 thf(fact_61_f11,axiom,
% 0.30/0.46 ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % f11
% 0.30/0.46 thf(fact_62_semiring_Osemiring__simprules_I11_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.30/0.46 = X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(11)
% 0.30/0.46 thf(fact_63_semiring_Osemiring__simprules_I6_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.30/0.46 = X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(6)
% 0.30/0.46 thf(fact_64_semiring_Ol__null,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.30/0.46 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.l_null
% 0.30/0.46 thf(fact_65_semiring_Or__null,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.30/0.46 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.r_null
% 0.30/0.46 thf(fact_66_semiring_Or__distr,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ Z @ ( add_a_b @ R3 @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.r_distr
% 0.30/0.46 thf(fact_67_semiring_Ol__distr,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.30/0.46 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.l_distr
% 0.30/0.46 thf(fact_68_add__pow__rdistr__int,axiom,
% 0.30/0.46 ! [A: a,B: a,K: int] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 0.30/0.46 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pow_rdistr_int
% 0.30/0.46 thf(fact_69_add__pow__ldistr__int,axiom,
% 0.30/0.46 ! [A: a,B: a,K: int] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 0.30/0.46 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pow_ldistr_int
% 0.30/0.46 thf(fact_70_inv__unique,axiom,
% 0.30/0.46 ! [Y: a,X: a,Y3: a] :
% 0.30/0.46 ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inv_unique
% 0.30/0.46 thf(fact_71_one__unique,axiom,
% 0.30/0.46 ! [U: a] :
% 0.30/0.46 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
% 0.30/0.46 = X3 ) )
% 0.30/0.46 => ( U
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % one_unique
% 0.30/0.46 thf(fact_72_add_Oint__pow__distrib,axiom,
% 0.30/0.46 ! [X: a,Y: a,I: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_distrib
% 0.30/0.46 thf(fact_73_add_Oint__pow__mult__distrib,axiom,
% 0.30/0.46 ! [X: a,Y: a,I: int] :
% 0.30/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( add_a_b @ r @ Y @ X ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_mult_distrib
% 0.30/0.46 thf(fact_74_semiring__simprules_I4_J,axiom,
% 0.30/0.46 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_simprules(4)
% 0.30/0.46 thf(fact_75_add_Oint__pow__closed,axiom,
% 0.30/0.46 ! [X: a,I: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_closed
% 0.30/0.46 thf(fact_76_add_Oint__pow__one,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_one
% 0.30/0.46 thf(fact_77_l__one,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_one
% 0.30/0.46 thf(fact_78_r__one,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_one
% 0.30/0.46 thf(fact_79_semiring_Osemiring__simprules_I4_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(4)
% 0.30/0.46 thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 0.30/0.46 = X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(9)
% 0.30/0.46 thf(fact_81_semiring_Osemiring__simprules_I3_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(3)
% 0.30/0.46 thf(fact_82_semiring_Osemiring__simprules_I8_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(8)
% 0.30/0.46 thf(fact_83_semiring_Osemiring__simprules_I1_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(1)
% 0.30/0.46 thf(fact_84_semiring_Osemiring__simprules_I5_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.30/0.46 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(5)
% 0.30/0.46 thf(fact_85_semiring_Osemiring__simprules_I7_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ X @ Y )
% 0.30/0.46 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(7)
% 0.30/0.46 thf(fact_86_semiring_Osemiring__simprules_I12_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) )
% 0.30/0.46 = ( add_a_b @ R3 @ Y @ ( add_a_b @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(12)
% 0.30/0.46 thf(fact_87_semiring_Osemiring__simprules_I2_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.semiring_simprules(2)
% 0.30/0.46 thf(fact_88_cring__fieldI2,axiom,
% 0.30/0.46 ( ( ( zero_a_b @ r )
% 0.30/0.46 != ( one_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ! [A4: a] :
% 0.30/0.46 ( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( A4
% 0.30/0.46 != ( zero_a_b @ r ) )
% 0.30/0.46 => ? [X5: a] :
% 0.30/0.46 ( ( member_a @ X5 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ r @ A4 @ X5 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) ) )
% 0.30/0.46 => ( field_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cring_fieldI2
% 0.30/0.46 thf(fact_89_add_Oint__pow__mult,axiom,
% 0.30/0.46 ! [X: a,I: int,J: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_mult
% 0.30/0.46 thf(fact_90_add__pow__ldistr,axiom,
% 0.30/0.46 ! [A: a,B: a,K: nat] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
% 0.30/0.46 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pow_ldistr
% 0.30/0.46 thf(fact_91_add__pow__rdistr,axiom,
% 0.30/0.46 ! [A: a,B: a,K: nat] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
% 0.30/0.46 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pow_rdistr
% 0.30/0.46 thf(fact_92_add_Ogroup__commutes__pow,axiom,
% 0.30/0.46 ! [X: a,Y: a,N: nat] :
% 0.30/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( add_a_b @ r @ Y @ X ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
% 0.30/0.46 = ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_commutes_pow
% 0.30/0.46 thf(fact_93_add_Opow__mult__distrib,axiom,
% 0.30/0.46 ! [X: a,Y: a,N: nat] :
% 0.30/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( add_a_b @ r @ Y @ X ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.pow_mult_distrib
% 0.30/0.46 thf(fact_94_add_Onat__pow__distrib,axiom,
% 0.30/0.46 ! [X: a,Y: a,N: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_distrib
% 0.30/0.46 thf(fact_95_add_Onat__pow__comm,axiom,
% 0.30/0.46 ! [X: a,N: nat,M: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_comm
% 0.30/0.46 thf(fact_96_add_Onat__pow__closed,axiom,
% 0.30/0.46 ! [X: a,N: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_closed
% 0.30/0.46 thf(fact_97_add_Onat__pow__one,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_one
% 0.30/0.46 thf(fact_98_one__not__zero,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( field_a_b @ R3 )
% 0.30/0.46 => ( ( one_a_ring_ext_a_b @ R3 )
% 0.30/0.46 != ( zero_a_b @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % one_not_zero
% 0.30/0.46 thf(fact_99_integral,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,A: a,B: a] :
% 0.30/0.46 ( ( field_a_b @ R3 )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
% 0.30/0.46 = ( zero_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( A
% 0.30/0.46 = ( zero_a_b @ R3 ) )
% 0.30/0.46 | ( B
% 0.30/0.46 = ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % integral
% 0.30/0.46 thf(fact_100_semiring_Oadd__pow__ldistr,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
% 0.30/0.46 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.add_pow_ldistr
% 0.30/0.46 thf(fact_101_semiring_Oadd__pow__rdistr,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.30/0.46 ( ( semiring_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
% 0.30/0.46 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring.add_pow_rdistr
% 0.30/0.46 thf(fact_102_add_Onat__pow__mult,axiom,
% 0.30/0.46 ! [X: a,N: nat,M: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 0.30/0.46 = ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_mult
% 0.30/0.46 thf(fact_103_add_Onat__pow__Suc2,axiom,
% 0.30/0.46 ! [X: a,N: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 0.30/0.46 = ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_Suc2
% 0.30/0.46 thf(fact_104_add_Opow__eq__div2,axiom,
% 0.30/0.46 ! [X: a,M: nat,N: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_pow_a_b_nat @ r @ M @ X )
% 0.30/0.46 = ( add_pow_a_b_nat @ r @ N @ X ) )
% 0.30/0.46 => ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.pow_eq_div2
% 0.30/0.46 thf(fact_105_minus__to__eq,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ r )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( a_minus_a_b @ r @ X @ Y )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( X = Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_to_eq
% 0.30/0.46 thf(fact_106_ring_Oequality,axiom,
% 0.30/0.46 ! [R: partia1833973666xt_a_b,R2: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( ( partia1066395285xt_a_b @ R )
% 0.30/0.46 = ( partia1066395285xt_a_b @ R2 ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ R )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ R2 ) )
% 0.30/0.46 => ( ( ( one_a_ring_ext_a_b @ R )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ R2 ) )
% 0.30/0.46 => ( ( ( zero_a_b @ R )
% 0.30/0.46 = ( zero_a_b @ R2 ) )
% 0.30/0.46 => ( ( ( add_a_b @ R )
% 0.30/0.46 = ( add_a_b @ R2 ) )
% 0.30/0.46 => ( ( ( more_a_b @ R )
% 0.30/0.46 = ( more_a_b @ R2 ) )
% 0.30/0.46 => ( R = R2 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ring.equality
% 0.30/0.46 thf(fact_107_is__abelian__group,axiom,
% 0.30/0.46 abelian_group_a_b @ r ).
% 0.30/0.46
% 0.30/0.46 % is_abelian_group
% 0.30/0.46 thf(fact_108_add_Onat__pow__Suc,axiom,
% 0.30/0.46 ! [N: nat,X: a] :
% 0.30/0.46 ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_Suc
% 0.30/0.46 thf(fact_109_abelian__groupE_I1_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(1)
% 0.30/0.46 thf(fact_110_abelian__groupE_I3_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.30/0.46 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(3)
% 0.30/0.46 thf(fact_111_abelian__groupE_I4_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ X @ Y )
% 0.30/0.46 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(4)
% 0.30/0.46 thf(fact_112_abelian__groupE_I2_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(2)
% 0.30/0.46 thf(fact_113_abelian__group_Ominus__closed,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_closed
% 0.30/0.46 thf(fact_114_abelian__groupI,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b] :
% 0.30/0.46 ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( member_a @ ( add_a_b @ R3 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) )
% 0.30/0.46 => ( ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ! [Z2: a] :
% 0.30/0.46 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X3 @ Y4 ) @ Z2 )
% 0.30/0.46 = ( add_a_b @ R3 @ X3 @ ( add_a_b @ R3 @ Y4 @ Z2 ) ) ) ) ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ X3 @ Y4 )
% 0.30/0.46 = ( add_a_b @ R3 @ Y4 @ X3 ) ) ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X3 )
% 0.30/0.46 = X3 ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ? [Xa: a] :
% 0.30/0.46 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 & ( ( add_a_b @ R3 @ Xa @ X3 )
% 0.30/0.46 = ( zero_a_b @ R3 ) ) ) )
% 0.30/0.46 => ( abelian_group_a_b @ R3 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupI
% 0.30/0.46 thf(fact_115_abelian__groupE_I5_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.30/0.46 = X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(5)
% 0.30/0.46 thf(fact_116_abelian__groupE_I6_J,axiom,
% 0.30/0.46 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ R3 )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.30/0.46 & ( ( add_a_b @ R3 @ X3 @ X )
% 0.30/0.46 = ( zero_a_b @ R3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_groupE(6)
% 0.30/0.46 thf(fact_117_abelian__group_Ofour__elem__comm,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a,D: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_minus_a_b @ G @ ( add_a_b @ G @ ( a_minus_a_b @ G @ A @ C ) @ B ) @ D )
% 0.30/0.46 = ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ ( add_a_b @ G @ A @ B ) @ C ) @ D ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.four_elem_comm
% 0.30/0.46 thf(fact_118_abelian__group_Oright__inv__add,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ C @ A ) @ B )
% 0.30/0.46 = ( a_minus_a_b @ G @ C @ ( add_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.right_inv_add
% 0.30/0.46 thf(fact_119_abelian__group_Ominus__to__eq,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( ( a_minus_a_b @ G @ X @ Y )
% 0.30/0.46 = ( zero_a_b @ G ) )
% 0.30/0.46 => ( X = Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_to_eq
% 0.30/0.46 thf(fact_120_diff__diff__left,axiom,
% 0.30/0.46 ! [I: nat,J: nat,K: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.46 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_left
% 0.30/0.46 thf(fact_121_Suc__diff__diff,axiom,
% 0.30/0.46 ! [M: nat,N: nat,K: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.30/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % Suc_diff_diff
% 0.30/0.46 thf(fact_122_diff__Suc__Suc,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_Suc_Suc
% 0.30/0.46 thf(fact_123_add__Suc__right,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.30/0.46 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_Suc_right
% 0.30/0.46 thf(fact_124_nat_Oinject,axiom,
% 0.30/0.46 ! [X2: nat,Y2: nat] :
% 0.30/0.46 ( ( ( suc @ X2 )
% 0.30/0.46 = ( suc @ Y2 ) )
% 0.30/0.46 = ( X2 = Y2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat.inject
% 0.30/0.46 thf(fact_125_old_Onat_Oinject,axiom,
% 0.30/0.46 ! [Nat: nat,Nat2: nat] :
% 0.30/0.46 ( ( ( suc @ Nat )
% 0.30/0.46 = ( suc @ Nat2 ) )
% 0.30/0.46 = ( Nat = Nat2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.nat.inject
% 0.30/0.46 thf(fact_126_Suc__inject,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( ( suc @ X )
% 0.30/0.46 = ( suc @ Y ) )
% 0.30/0.46 => ( X = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % Suc_inject
% 0.30/0.46 thf(fact_127_n__not__Suc__n,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( N
% 0.30/0.46 != ( suc @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % n_not_Suc_n
% 0.30/0.46 thf(fact_128_diff__commute,axiom,
% 0.30/0.46 ! [I: nat,J: nat,K: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_commute
% 0.30/0.46 thf(fact_129_add__Suc,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.46 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_Suc
% 0.30/0.46 thf(fact_130_nat__arith_Osuc1,axiom,
% 0.30/0.46 ! [A3: nat,K: nat,A: nat] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.46 => ( ( suc @ A3 )
% 0.30/0.46 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_arith.suc1
% 0.30/0.46 thf(fact_131_add__Suc__shift,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.46 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_Suc_shift
% 0.30/0.46 thf(fact_132_zero__induct__lemma,axiom,
% 0.30/0.46 ! [P: nat > $o,K: nat,I: nat] :
% 0.30/0.46 ( ( P @ K )
% 0.30/0.46 => ( ! [N2: nat] :
% 0.30/0.46 ( ( P @ ( suc @ N2 ) )
% 0.30/0.46 => ( P @ N2 ) )
% 0.30/0.46 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_induct_lemma
% 0.30/0.46 thf(fact_133_diff__add__inverse2,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.30/0.46 = M ) ).
% 0.30/0.46
% 0.30/0.46 % diff_add_inverse2
% 0.30/0.46 thf(fact_134_diff__add__inverse,axiom,
% 0.30/0.46 ! [N: nat,M: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.30/0.46 = M ) ).
% 0.30/0.46
% 0.30/0.46 % diff_add_inverse
% 0.30/0.46 thf(fact_135_diff__cancel2,axiom,
% 0.30/0.46 ! [M: nat,K: nat,N: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_cancel2
% 0.30/0.46 thf(fact_136_Nat_Odiff__cancel,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nat.diff_cancel
% 0.30/0.46 thf(fact_137_add__diff__cancel__right_H,axiom,
% 0.30/0.46 ! [A: nat,B: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_right'
% 0.30/0.46 thf(fact_138_add__diff__cancel__right_H,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_right'
% 0.30/0.46 thf(fact_139_add__diff__cancel__right,axiom,
% 0.30/0.46 ! [A: nat,C: nat,B: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.46 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_right
% 0.30/0.46 thf(fact_140_add__diff__cancel__right,axiom,
% 0.30/0.46 ! [A: int,C: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_right
% 0.30/0.46 thf(fact_141_add__diff__cancel__left_H,axiom,
% 0.30/0.46 ! [A: nat,B: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.30/0.46 = B ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_left'
% 0.30/0.46 thf(fact_142_add__diff__cancel__left_H,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.30/0.46 = B ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_left'
% 0.30/0.46 thf(fact_143_add__diff__cancel__left,axiom,
% 0.30/0.46 ! [C: nat,A: nat,B: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.46 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_left
% 0.30/0.46 thf(fact_144_add__diff__cancel__left,axiom,
% 0.30/0.46 ! [C: int,A: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel_left
% 0.30/0.46 thf(fact_145_add__right__cancel,axiom,
% 0.30/0.46 ! [B: int,A: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.46 = ( plus_plus_int @ C @ A ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_cancel
% 0.30/0.46 thf(fact_146_add__right__cancel,axiom,
% 0.30/0.46 ! [B: nat,A: nat,C: nat] :
% 0.30/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_cancel
% 0.30/0.46 thf(fact_147_add__left__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.46 = ( plus_plus_int @ A @ C ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_cancel
% 0.30/0.46 thf(fact_148_add__left__cancel,axiom,
% 0.30/0.46 ! [A: nat,B: nat,C: nat] :
% 0.30/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_cancel
% 0.30/0.46 thf(fact_149_add__diff__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_cancel
% 0.30/0.46 thf(fact_150_diff__add__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % diff_add_cancel
% 0.30/0.46 thf(fact_151_add__right__imp__eq,axiom,
% 0.30/0.46 ! [B: int,A: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.46 = ( plus_plus_int @ C @ A ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_imp_eq
% 0.30/0.46 thf(fact_152_add__right__imp__eq,axiom,
% 0.30/0.46 ! [B: nat,A: nat,C: nat] :
% 0.30/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_imp_eq
% 0.30/0.46 thf(fact_153_add__left__imp__eq,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.46 = ( plus_plus_int @ A @ C ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_imp_eq
% 0.30/0.46 thf(fact_154_add__left__imp__eq,axiom,
% 0.30/0.46 ! [A: nat,B: nat,C: nat] :
% 0.30/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_imp_eq
% 0.30/0.46 thf(fact_155_add_Oleft__commute,axiom,
% 0.30/0.46 ! [B: int,A: int,C: int] :
% 0.30/0.46 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.30/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_commute
% 0.30/0.46 thf(fact_156_add_Oleft__commute,axiom,
% 0.30/0.46 ! [B: nat,A: nat,C: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.30/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_commute
% 0.30/0.46 thf(fact_157_add_Ocommute,axiom,
% 0.30/0.46 ( plus_plus_int
% 0.30/0.46 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.commute
% 0.30/0.46 thf(fact_158_add_Ocommute,axiom,
% 0.30/0.46 ( plus_plus_nat
% 0.30/0.46 = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.commute
% 0.30/0.46 thf(fact_159_group__add__class_Oadd_Oright__cancel,axiom,
% 0.30/0.46 ! [B: int,A: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.46 = ( plus_plus_int @ C @ A ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_add_class.add.right_cancel
% 0.30/0.46 thf(fact_160_add_Oleft__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.46 = ( plus_plus_int @ A @ C ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_cancel
% 0.30/0.46 thf(fact_161_add_Oassoc,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.assoc
% 0.30/0.46 thf(fact_162_add_Oassoc,axiom,
% 0.30/0.46 ! [A: nat,B: nat,C: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.assoc
% 0.30/0.46 thf(fact_163_group__cancel_Oadd2,axiom,
% 0.30/0.46 ! [B5: int,K: int,B: int,A: int] :
% 0.30/0.46 ( ( B5
% 0.30/0.46 = ( plus_plus_int @ K @ B ) )
% 0.30/0.46 => ( ( plus_plus_int @ A @ B5 )
% 0.30/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add2
% 0.30/0.46 thf(fact_164_group__cancel_Oadd2,axiom,
% 0.30/0.46 ! [B5: nat,K: nat,B: nat,A: nat] :
% 0.30/0.46 ( ( B5
% 0.30/0.46 = ( plus_plus_nat @ K @ B ) )
% 0.30/0.46 => ( ( plus_plus_nat @ A @ B5 )
% 0.30/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add2
% 0.30/0.46 thf(fact_165_group__cancel_Oadd1,axiom,
% 0.30/0.46 ! [A3: int,K: int,A: int,B: int] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_int @ K @ A ) )
% 0.30/0.46 => ( ( plus_plus_int @ A3 @ B )
% 0.30/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add1
% 0.30/0.46 thf(fact_166_group__cancel_Oadd1,axiom,
% 0.30/0.46 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.46 => ( ( plus_plus_nat @ A3 @ B )
% 0.30/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add1
% 0.30/0.46 thf(fact_167_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ( plus_plus_int @ I @ K )
% 0.30/0.46 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(4)
% 0.30/0.46 thf(fact_168_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ( plus_plus_nat @ I @ K )
% 0.30/0.46 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(4)
% 0.30/0.46 thf(fact_169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.46 thf(fact_170_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.46 ! [A: nat,B: nat,C: nat] :
% 0.30/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.46 thf(fact_171_diff__right__commute,axiom,
% 0.30/0.46 ! [A: nat,C: nat,B: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.30/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_commute
% 0.30/0.46 thf(fact_172_diff__right__commute,axiom,
% 0.30/0.46 ! [A: int,C: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.30/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_commute
% 0.30/0.46 thf(fact_173_diff__eq__diff__eq,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.46 = ( minus_minus_int @ C @ D ) )
% 0.30/0.46 => ( ( A = B )
% 0.30/0.46 = ( C = D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_eq
% 0.30/0.46 thf(fact_174_group__cancel_Osub1,axiom,
% 0.30/0.46 ! [A3: int,K: int,A: int,B: int] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_int @ K @ A ) )
% 0.30/0.46 => ( ( minus_minus_int @ A3 @ B )
% 0.30/0.46 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.sub1
% 0.30/0.46 thf(fact_175_diff__eq__eq,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.46 = C )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_eq
% 0.30/0.46 thf(fact_176_eq__diff__eq,axiom,
% 0.30/0.46 ! [A: int,C: int,B: int] :
% 0.30/0.46 ( ( A
% 0.30/0.46 = ( minus_minus_int @ C @ B ) )
% 0.30/0.46 = ( ( plus_plus_int @ A @ B )
% 0.30/0.46 = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_diff_eq
% 0.30/0.46 thf(fact_177_add__diff__eq,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_diff_eq
% 0.30/0.46 thf(fact_178_diff__diff__eq2,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_eq2
% 0.30/0.46 thf(fact_179_diff__add__eq,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_add_eq
% 0.30/0.46 thf(fact_180_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_add_eq_diff_diff_swap
% 0.30/0.46 thf(fact_181_diff__diff__add,axiom,
% 0.30/0.46 ! [A: nat,B: nat,C: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.30/0.46 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_add
% 0.30/0.46 thf(fact_182_diff__diff__add,axiom,
% 0.30/0.46 ! [A: int,B: int,C: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.46 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_add
% 0.30/0.46 thf(fact_183_add__implies__diff,axiom,
% 0.30/0.46 ! [C: nat,B: nat,A: nat] :
% 0.30/0.46 ( ( ( plus_plus_nat @ C @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( C
% 0.30/0.46 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_implies_diff
% 0.30/0.46 thf(fact_184_add__implies__diff,axiom,
% 0.30/0.46 ! [C: int,B: int,A: int] :
% 0.30/0.46 ( ( ( plus_plus_int @ C @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( C
% 0.30/0.46 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_implies_diff
% 0.30/0.46 thf(fact_185_Units__l__inv__ex,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_l_inv_ex
% 0.30/0.46 thf(fact_186_Units__r__inv__ex,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_r_inv_ex
% 0.30/0.46 thf(fact_187_group__l__invI,axiom,
% 0.30/0.46 ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ? [Xa: a] :
% 0.30/0.46 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ r @ Xa @ X3 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) )
% 0.30/0.46 => ( group_a_ring_ext_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_l_invI
% 0.30/0.46 thf(fact_188_Units__closed,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_closed
% 0.30/0.46 thf(fact_189_Units__inv__comm,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_inv_comm
% 0.30/0.46 thf(fact_190_Units__m__closed,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_m_closed
% 0.30/0.46 thf(fact_191_Units__one__closed,axiom,
% 0.30/0.46 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % Units_one_closed
% 0.30/0.46 thf(fact_192_Units__l__cancel,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
% 0.30/0.46 = ( Y = Z ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Units_l_cancel
% 0.30/0.46 thf(fact_193_groupI,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit] :
% 0.30/0.46 ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ ( partia1955795460t_unit @ G ) ) ) )
% 0.30/0.46 => ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ! [Z2: a] :
% 0.30/0.46 ( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ Z2 )
% 0.30/0.46 = ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y4 @ Z2 ) ) ) ) ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X3 )
% 0.30/0.46 = X3 ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ? [Xa: a] :
% 0.30/0.46 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 & ( ( mult_a_Product_unit @ G @ Xa @ X3 )
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) )
% 0.30/0.46 => ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % groupI
% 0.30/0.46 thf(fact_194_groupI,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b] :
% 0.30/0.46 ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 0.30/0.46 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ! [Y4: a] :
% 0.30/0.46 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ! [Z2: a] :
% 0.30/0.46 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ Z2 )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y4 @ Z2 ) ) ) ) ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X3 )
% 0.30/0.46 = X3 ) )
% 0.30/0.46 => ( ! [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ? [Xa: a] :
% 0.30/0.46 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X3 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 0.30/0.46 => ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % groupI
% 0.30/0.46 thf(fact_195_group_Oinv__comm,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( ( mult_a_Product_unit @ G @ X @ Y )
% 0.30/0.46 = ( one_a_Product_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.inv_comm
% 0.30/0.46 thf(fact_196_group_Oinv__comm,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.inv_comm
% 0.30/0.46 thf(fact_197_group_Ol__inv__ex,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 & ( ( mult_a_Product_unit @ G @ X3 @ X )
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_inv_ex
% 0.30/0.46 thf(fact_198_group_Ol__inv__ex,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ G @ X3 @ X )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_inv_ex
% 0.30/0.46 thf(fact_199_Group_Ogroup_Oright__cancel,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.30/0.46 = ( mult_a_Product_unit @ G @ Z @ X ) )
% 0.30/0.46 = ( Y = Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Group.group.right_cancel
% 0.30/0.46 thf(fact_200_Group_Ogroup_Oright__cancel,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
% 0.30/0.46 = ( Y = Z ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Group.group.right_cancel
% 0.30/0.46 thf(fact_201_group_OUnits__eq,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( units_a_Product_unit @ G )
% 0.30/0.46 = ( partia1955795460t_unit @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.Units_eq
% 0.30/0.46 thf(fact_202_group_OUnits__eq,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( units_a_ring_ext_a_b @ G )
% 0.30/0.46 = ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.Units_eq
% 0.30/0.46 thf(fact_203_group_Or__cancel__one_H,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( mult_a_Product_unit @ G @ A @ X ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_cancel_one'
% 0.30/0.46 thf(fact_204_group_Or__cancel__one_H,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ A @ X ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_cancel_one'
% 0.30/0.46 thf(fact_205_group_Ol__cancel__one_H,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( mult_a_Product_unit @ G @ X @ A ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_cancel_one'
% 0.30/0.46 thf(fact_206_group_Ol__cancel__one_H,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( X
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ X @ A ) )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_cancel_one'
% 0.30/0.46 thf(fact_207_group_Or__cancel__one,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( ( mult_a_Product_unit @ G @ A @ X )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_cancel_one
% 0.30/0.46 thf(fact_208_group_Or__cancel__one,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_cancel_one
% 0.30/0.46 thf(fact_209_group_Ol__cancel__one,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( ( mult_a_Product_unit @ G @ X @ A )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_cancel_one
% 0.30/0.46 thf(fact_210_group_Ol__cancel__one,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
% 0.30/0.46 = X )
% 0.30/0.46 = ( A
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.l_cancel_one
% 0.30/0.46 thf(fact_211_group_Or__inv__ex,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a] :
% 0.30/0.46 ( ( group_a_Product_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 & ( ( mult_a_Product_unit @ G @ X @ X3 )
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_inv_ex
% 0.30/0.46 thf(fact_212_group_Or__inv__ex,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( group_a_ring_ext_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ? [X3: a] :
% 0.30/0.46 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 & ( ( mult_a_ring_ext_a_b @ G @ X @ X3 )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group.r_inv_ex
% 0.30/0.46 thf(fact_213_units__group,axiom,
% 0.30/0.46 group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % units_group
% 0.30/0.46 thf(fact_214_inv__add,axiom,
% 0.30/0.46 ! [A: a,B: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 0.30/0.46 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inv_add
% 0.30/0.46 thf(fact_215_r__neg2,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 0.30/0.46 = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_neg2
% 0.30/0.46 thf(fact_216_r__neg1,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_neg1
% 0.30/0.46 thf(fact_217_local_Ominus__add,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.minus_add
% 0.30/0.46 thf(fact_218_add_Oinv__solve__right_H,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 0.30/0.46 = A )
% 0.30/0.46 = ( B
% 0.30/0.46 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_solve_right'
% 0.30/0.46 thf(fact_219_add_Oinv__solve__right,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( A
% 0.30/0.46 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 0.30/0.46 = ( B
% 0.30/0.46 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_solve_right
% 0.30/0.46 thf(fact_220_add_Oinv__solve__left_H,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 0.30/0.46 = A )
% 0.30/0.46 = ( C
% 0.30/0.46 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_solve_left'
% 0.30/0.46 thf(fact_221_add_Oinv__solve__left,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( A
% 0.30/0.46 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 0.30/0.46 = ( C
% 0.30/0.46 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_solve_left
% 0.30/0.46 thf(fact_222_add_Oinv__mult__group,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_mult_group
% 0.30/0.46 thf(fact_223_a__transpose__inv,axiom,
% 0.30/0.46 ! [X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = Z )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 0.30/0.46 = Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % a_transpose_inv
% 0.30/0.46 thf(fact_224_r__minus,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_minus
% 0.30/0.46 thf(fact_225_l__minus,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_minus
% 0.30/0.46 thf(fact_226_add_Onat__pow__inv,axiom,
% 0.30/0.46 ! [X: a,I: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.nat_pow_inv
% 0.30/0.46 thf(fact_227_add_Oint__pow__inv,axiom,
% 0.30/0.46 ! [X: a,I: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_inv
% 0.30/0.46 thf(fact_228_minus__eq,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( a_minus_a_b @ r @ X @ Y )
% 0.30/0.46 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_eq
% 0.30/0.46 thf(fact_229_sum__zero__eq__neg,axiom,
% 0.30/0.46 ! [X: a,Y: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( add_a_b @ r @ X @ Y )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( X
% 0.30/0.46 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_zero_eq_neg
% 0.30/0.46 thf(fact_230_r__neg,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % r_neg
% 0.30/0.46 thf(fact_231_minus__equality,axiom,
% 0.30/0.46 ! [Y: a,X: a] :
% 0.30/0.46 ( ( ( add_a_b @ r @ Y @ X )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_inv_a_b @ r @ X )
% 0.30/0.46 = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_equality
% 0.30/0.46 thf(fact_232_l__neg,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % l_neg
% 0.30/0.46 thf(fact_233_add_Oint__pow__diff,axiom,
% 0.30/0.46 ! [X: a,N: int,M: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
% 0.30/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_diff
% 0.30/0.46 thf(fact_234_local_Ominus__minus,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.minus_minus
% 0.30/0.46 thf(fact_235_add_Oinv__closed,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_closed
% 0.30/0.46 thf(fact_236_local_Ominus__zero,axiom,
% 0.30/0.46 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 0.30/0.46 = ( zero_a_b @ r ) ) ).
% 0.30/0.46
% 0.30/0.46 % local.minus_zero
% 0.30/0.46 thf(fact_237_add_Oinv__eq__1__iff,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( ( a_inv_a_b @ r @ X )
% 0.30/0.46 = ( zero_a_b @ r ) )
% 0.30/0.46 = ( X
% 0.30/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inv_eq_1_iff
% 0.30/0.46 thf(fact_238_Units__minus__one__closed,axiom,
% 0.30/0.46 member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % Units_minus_one_closed
% 0.30/0.46 thf(fact_239_units__of__mult,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit] :
% 0.30/0.46 ( ( mult_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 0.30/0.46 = ( mult_a_Product_unit @ G ) ) ).
% 0.30/0.46
% 0.30/0.46 % units_of_mult
% 0.30/0.46 thf(fact_240_units__of__mult,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G ) ) ).
% 0.30/0.46
% 0.30/0.46 % units_of_mult
% 0.30/0.46 thf(fact_241_abelian__group_Oa__inv__closed,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.a_inv_closed
% 0.30/0.46 thf(fact_242_abelian__group_Ominus__minus,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
% 0.30/0.46 = X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_minus
% 0.30/0.46 thf(fact_243_a__minus__def,axiom,
% 0.30/0.46 ( a_minus_a_b
% 0.30/0.46 = ( ^ [R4: partia1833973666xt_a_b,X4: a,Y5: a] : ( add_a_b @ R4 @ X4 @ ( a_inv_a_b @ R4 @ Y5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % a_minus_def
% 0.30/0.46 thf(fact_244_units__of__one,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit] :
% 0.30/0.46 ( ( one_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 0.30/0.46 = ( one_a_Product_unit @ G ) ) ).
% 0.30/0.46
% 0.30/0.46 % units_of_one
% 0.30/0.46 thf(fact_245_units__of__one,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.30/0.46 = ( one_a_ring_ext_a_b @ G ) ) ).
% 0.30/0.46
% 0.30/0.46 % units_of_one
% 0.30/0.46 thf(fact_246_abelian__group_Ominus__add,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
% 0.30/0.46 = ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_add
% 0.30/0.46 thf(fact_247_abelian__group_Or__neg2,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
% 0.30/0.46 = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.r_neg2
% 0.30/0.46 thf(fact_248_abelian__group_Or__neg1,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
% 0.30/0.46 = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.r_neg1
% 0.30/0.46 thf(fact_249_abelian__group_Ominus__eq,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( a_minus_a_b @ G @ X @ Y )
% 0.30/0.46 = ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_eq
% 0.30/0.46 thf(fact_250_abelian__group_Ol__neg,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
% 0.30/0.46 = ( zero_a_b @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.l_neg
% 0.30/0.46 thf(fact_251_abelian__group_Or__neg,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
% 0.30/0.46 = ( zero_a_b @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.r_neg
% 0.30/0.46 thf(fact_252_abelian__group_Ominus__equality,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,Y: a,X: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( ( add_a_b @ G @ Y @ X )
% 0.30/0.46 = ( zero_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_inv_a_b @ G @ X )
% 0.30/0.46 = Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.minus_equality
% 0.30/0.46 thf(fact_253_abelian__group_Oinv__add,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,A: a,B: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ A @ B ) )
% 0.30/0.46 = ( a_minus_a_b @ G @ ( a_inv_a_b @ G @ A ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.inv_add
% 0.30/0.46 thf(fact_254_abelian__group_Oa__transpose__inv,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( abelian_group_a_b @ G )
% 0.30/0.46 => ( ( ( add_a_b @ G @ X @ Y )
% 0.30/0.46 = Z )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Z )
% 0.30/0.46 = Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abelian_group.a_transpose_inv
% 0.30/0.46 thf(fact_255_add_Oint__pow__neg,axiom,
% 0.30/0.46 ! [X: a,I: int] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_neg
% 0.30/0.46 thf(fact_256_add__minus__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.30/0.46 = B ) ).
% 0.30/0.46
% 0.30/0.46 % add_minus_cancel
% 0.30/0.46 thf(fact_257_minus__add__cancel,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.30/0.46 = B ) ).
% 0.30/0.46
% 0.30/0.46 % minus_add_cancel
% 0.30/0.46 thf(fact_258_minus__add__distrib,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_add_distrib
% 0.30/0.46 thf(fact_259_minus__diff__eq,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.46 = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_diff_eq
% 0.30/0.46 thf(fact_260_diff__minus__eq__add,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.46 = ( plus_plus_int @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_minus_eq_add
% 0.30/0.46 thf(fact_261_uminus__add__conv__diff,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.46 = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_add_conv_diff
% 0.30/0.46 thf(fact_262_group__cancel_Osub2,axiom,
% 0.30/0.46 ! [B5: int,K: int,B: int,A: int] :
% 0.30/0.46 ( ( B5
% 0.30/0.46 = ( plus_plus_int @ K @ B ) )
% 0.30/0.46 => ( ( minus_minus_int @ A @ B5 )
% 0.30/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.sub2
% 0.30/0.46 thf(fact_263_diff__conv__add__uminus,axiom,
% 0.30/0.46 ( minus_minus_int
% 0.30/0.46 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_conv_add_uminus
% 0.30/0.46 thf(fact_264_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.46 ( minus_minus_int
% 0.30/0.46 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.46 thf(fact_265_group__cancel_Oneg1,axiom,
% 0.30/0.46 ! [A3: int,K: int,A: int] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_int @ K @ A ) )
% 0.30/0.46 => ( ( uminus_uminus_int @ A3 )
% 0.30/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.neg1
% 0.30/0.46 thf(fact_266_add_Oinverse__distrib__swap,axiom,
% 0.30/0.46 ! [A: int,B: int] :
% 0.30/0.46 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.inverse_distrib_swap
% 0.30/0.46 thf(fact_267_minus__diff__commute,axiom,
% 0.30/0.46 ! [B: int,A: int] :
% 0.30/0.46 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.30/0.46 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_diff_commute
% 0.30/0.46 thf(fact_268_add_Oint__pow__neg__int,axiom,
% 0.30/0.46 ! [X: a,N: nat] :
% 0.30/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.30/0.46 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) @ X )
% 0.30/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.int_pow_neg_int
% 0.30/0.46 thf(fact_269_units__comm__group,axiom,
% 0.30/0.46 comm_g1684316527t_unit @ ( units_1411277569xt_a_b @ r ) ).
% 0.30/0.46
% 0.30/0.46 % units_comm_group
% 0.30/0.46 thf(fact_270_of__nat__eq__iff,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( ( semiri1382578993at_nat @ M )
% 0.30/0.46 = ( semiri1382578993at_nat @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_nat_eq_iff
% 0.30/0.46 thf(fact_271_of__nat__eq__iff,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.46 = ( semiri2019852685at_int @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_nat_eq_iff
% 0.30/0.46 thf(fact_272_of__nat__add,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( semiri1382578993at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.46 = ( plus_plus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_nat_add
% 0.30/0.46 thf(fact_273_of__nat__add,axiom,
% 0.30/0.46 ! [M: nat,N: nat] :
% 0.30/0.46 ( ( semiri2019852685at_int @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.46 = ( plus_plus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_nat_add
% 0.30/0.46 thf(fact_274_comm__groupE_I4_J,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.30/0.46 ( ( comm_g1684316527t_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( mult_a_Product_unit @ G @ X @ Y )
% 0.30/0.46 = ( mult_a_Product_unit @ G @ Y @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(4)
% 0.30/0.46 thf(fact_275_comm__groupE_I4_J,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( comm_g791708116xt_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ Y @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(4)
% 0.30/0.46 thf(fact_276_comm__groupE_I3_J,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( comm_g1684316527t_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ Z )
% 0.30/0.46 = ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(3)
% 0.30/0.46 thf(fact_277_comm__groupE_I3_J,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.30/0.46 ( ( comm_g791708116xt_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
% 0.30/0.46 = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(3)
% 0.30/0.46 thf(fact_278_comm__groupE_I1_J,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.30/0.46 ( ( comm_g1684316527t_unit @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.30/0.46 => ( member_a @ ( mult_a_Product_unit @ G @ X @ Y ) @ ( partia1955795460t_unit @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(1)
% 0.30/0.46 thf(fact_279_comm__groupE_I1_J,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.30/0.46 ( ( comm_g791708116xt_a_b @ G )
% 0.30/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.30/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(1)
% 0.30/0.46 thf(fact_280_comm__groupE_I2_J,axiom,
% 0.30/0.46 ! [G: partia96731725t_unit] :
% 0.30/0.46 ( ( comm_g1684316527t_unit @ G )
% 0.30/0.46 => ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(2)
% 0.30/0.46 thf(fact_281_comm__groupE_I2_J,axiom,
% 0.30/0.46 ! [G: partia1833973666xt_a_b] :
% 0.30/0.46 ( ( comm_g791708116xt_a_b @ G )
% 0.30/0.46 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comm_groupE(2)
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.73/1.59 ---- Embedded in THF ---
% 2.73/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.73/1.59 %%% Generated on Tue Feb 24 08:26:01 EST 2026
% 2.73/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.73/1.59 %%% Logic specification used:
% 2.73/1.59 %%% thf(spec, logic, $$dhol).
% 2.73/1.59
% 2.73/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2DTF4744.p
% See solution above
% 2.73/1.59 ------------------------
% 3.05/1.61 ---- Cleaned THF ---
% 3.05/1.62 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J_J,type,
% 3.05/1.62 partia1993116613t_unit: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
% 3.05/1.62 partia1833973666xt_a_b: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
% 3.05/1.62 partia96731725t_unit: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.05/1.62 set_Product_prod_a_a: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.05/1.62 product_prod_a_a: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.05/1.62 set_a: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Nat__Onat,type,
% 3.05/1.62 nat: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_t__Int__Oint,type,
% 3.05/1.62 int: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_tf__b,type,
% 3.05/1.62 b: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(ty_n_tf__a,type,
% 3.05/1.62 a: $tType ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
% 3.05/1.62 partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
% 3.05/1.62 partia1955795460t_unit: partia96731725t_unit > set_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
% 3.05/1.62 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 units_a_Product_unit: partia96731725t_unit > set_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 comm_g1684316527t_unit: partia96731725t_unit > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 comm_g791708116xt_a_b: partia1833973666xt_a_b > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 group_a_Product_unit: partia96731725t_unit > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 mult_a_Product_unit: partia96731725t_unit > a > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 one_a_Product_unit: partia96731725t_unit > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ounits__of_001tf__a_001t__Product____Type__Ounit,type,
% 3.05/1.62 units_873712258t_unit: partia96731725t_unit > partia96731725t_unit ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 3.05/1.62 units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 3.05/1.62 minus_minus_int: int > int > int ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 3.05/1.62 minus_minus_nat: nat > nat > nat ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 3.05/1.62 plus_plus_int: int > int > int ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 3.05/1.62 plus_plus_nat: nat > nat > nat ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 3.05/1.62 uminus_uminus_int: int > int ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Nat_OSuc,type,
% 3.05/1.62 suc: nat > nat ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 3.05/1.62 semiri2019852685at_int: nat > int ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 3.05/1.62 semiri1382578993at_nat: nat > nat ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 3.05/1.62 product_Pair_a_a: a > a > product_prod_a_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 3.05/1.62 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 3.05/1.62 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
% 3.05/1.62 abelian_group_a_b: partia1833973666xt_a_b > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 3.05/1.62 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
% 3.05/1.62 add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
% 3.05/1.62 field_a_b: partia1833973666xt_a_b > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 3.05/1.62 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oring_Omore_001tf__a_001tf__b,type,
% 3.05/1.62 more_a_b: partia1833973666xt_a_b > b ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 3.05/1.62 zero_a_b: partia1833973666xt_a_b > a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 3.05/1.62 semiring_a_b: partia1833973666xt_a_b > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.05/1.62 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.05/1.62 collect_a: ( a > $o ) > set_a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.05/1.62 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_c_member_001tf__a,type,
% 3.05/1.62 member_a: a > set_a > $o ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_R,type,
% 3.05/1.62 r: partia1833973666xt_a_b ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_r,type,
% 3.05/1.62 r2: a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_r_H,type,
% 3.05/1.62 r3: a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_r_H_H,type,
% 3.05/1.62 r4: a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_rel,type,
% 3.05/1.62 rel: partia1993116613t_unit ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_s,type,
% 3.05/1.62 s: a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_s_H,type,
% 3.05/1.62 s2: a ).
% 3.05/1.62
% 3.05/1.62 thf(sy_v_s_H_H,type,
% 3.05/1.62 s3: a ).
% 3.05/1.62
% 3.05/1.62 thf(fact_0_right__add__eq,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( A = B )
% 3.05/1.62 => ( ( add_a_b @ r @ C @ A )
% 3.05/1.62 = ( add_a_b @ r @ C @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_1_assms_I3_J,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ r4 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_2_assms_I2_J,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_3_assms_I1_J,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_4_closed__rel__add,axiom,
% 3.05/1.62 ! [R: a,S: a,R2: a,S2: a] :
% 3.05/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.05/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 3.05/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ S2 @ R ) @ ( mult_a_ring_ext_a_b @ r @ S @ R2 ) ) @ ( mult_a_ring_ext_a_b @ r @ S @ S2 ) ) @ ( partia206007992t_unit @ rel ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) @ s3 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_6_f9,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ s ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_9_f10,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_10_f5,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_11_f4,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_14_f7,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
% 3.05/1.62 = ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_15_f8,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) )
% 3.05/1.62 = ( 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 @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_16_f12,axiom,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_17_l__distr,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 3.05/1.62 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_18_r__distr,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062,axiom,
% 3.05/1.62 ( ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_20_prod_Oinject,axiom,
% 3.05/1.62 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 3.05/1.62 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 3.05/1.62 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 3.05/1.62 = ( ( X1 = Y1 )
% 3.05/1.62 & ( X2 = Y2 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_21_old_Oprod_Oinject,axiom,
% 3.05/1.62 ! [A: a,B: a,A2: a,B2: a] :
% 3.05/1.62 ( ( ( product_Pair_a_a @ A @ B )
% 3.05/1.62 = ( product_Pair_a_a @ A2 @ B2 ) )
% 3.05/1.62 = ( ( A = A2 )
% 3.05/1.62 & ( B = B2 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_22_local_Osemiring__axioms,axiom,
% 3.05/1.62 semiring_a_b @ r ).
% 3.05/1.62
% 3.05/1.62 thf(fact_23_m__assoc,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_24_m__comm,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_25_m__lcomm,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_26_add_Om__assoc,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 3.05/1.62 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_27_add_Om__comm,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_28_add_Om__lcomm,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 3.05/1.62 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_29_local_Oright__minus__eq,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( A = B )
% 3.05/1.62 => ( ( a_minus_a_b @ r @ C @ A )
% 3.05/1.62 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_30_local_Ominus__unique,axiom,
% 3.05/1.62 ! [Y: a,X: a,Y3: a] :
% 3.05/1.62 ( ( ( add_a_b @ r @ Y @ X )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ X @ Y3 )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( Y = Y3 ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_31_add_Or__inv__ex,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( add_a_b @ r @ X @ X3 )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_32_add_Oone__unique,axiom,
% 3.05/1.62 ! [U: a] :
% 3.05/1.62 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ U @ X3 )
% 3.05/1.62 = X3 ) )
% 3.05/1.62 => ( U
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_33_add_Ol__inv__ex,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( add_a_b @ r @ X3 @ X )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_34_add_Oinv__comm,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ Y @ X )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_35_right__inv__add,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 3.05/1.62 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_36_four__elem__comm,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a,D: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 3.05/1.62 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_37_local_Oadd_Oright__cancel,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ Y @ X )
% 3.05/1.62 = ( add_a_b @ r @ Z @ X ) )
% 3.05/1.62 = ( Y = Z ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_38_add_Om__closed,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_39_zero__closed,axiom,
% 3.05/1.62 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_40_semiring__simprules_I3_J,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_41_minus__closed,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_42_r__zero,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 3.05/1.62 = X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_43_l__zero,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 3.05/1.62 = X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_44_add_Or__cancel__one_H,axiom,
% 3.05/1.62 ! [X: a,A: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( add_a_b @ r @ A @ X ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_45_mem__Collect__eq,axiom,
% 3.05/1.62 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 3.05/1.62 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 3.05/1.62 = ( P @ A ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_46_mem__Collect__eq,axiom,
% 3.05/1.62 ! [A: a,P: a > $o] :
% 3.05/1.62 ( ( member_a @ A @ ( collect_a @ P ) )
% 3.05/1.62 = ( P @ A ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_47_Collect__mem__eq,axiom,
% 3.05/1.62 ! [A3: set_Product_prod_a_a] :
% 3.05/1.62 ( ( collec645855634od_a_a
% 3.05/1.62 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A3 ) )
% 3.05/1.62 = A3 ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_48_Collect__mem__eq,axiom,
% 3.05/1.62 ! [A3: set_a] :
% 3.05/1.62 ( ( collect_a
% 3.05/1.62 @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
% 3.05/1.62 = A3 ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_49_Collect__cong,axiom,
% 3.05/1.62 ! [P: a > $o,Q: a > $o] :
% 3.05/1.62 ( ! [X3: a] :
% 3.05/1.62 ( ( P @ X3 )
% 3.05/1.62 = ( Q @ X3 ) )
% 3.05/1.62 => ( ( collect_a @ P )
% 3.05/1.62 = ( collect_a @ Q ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_50_Collect__cong,axiom,
% 3.05/1.62 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.05/1.62 ( ! [X3: product_prod_a_a] :
% 3.05/1.62 ( ( P @ X3 )
% 3.05/1.62 = ( Q @ X3 ) )
% 3.05/1.62 => ( ( collec645855634od_a_a @ P )
% 3.05/1.62 = ( collec645855634od_a_a @ Q ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_51_add_Or__cancel__one,axiom,
% 3.05/1.62 ! [X: a,A: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ A @ X )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_52_add_Ol__cancel__one_H,axiom,
% 3.05/1.62 ! [X: a,A: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( add_a_b @ r @ X @ A ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_53_add_Ol__cancel__one,axiom,
% 3.05/1.62 ! [X: a,A: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ X @ A )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_54_r__null,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_55_l__null,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_56_old_Oprod_Oinducts,axiom,
% 3.05/1.62 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 3.05/1.62 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 3.05/1.62 => ( P @ Prod ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_57_old_Oprod_Oexhaust,axiom,
% 3.05/1.62 ! [Y: product_prod_a_a] :
% 3.05/1.62 ~ ! [A4: a,B3: a] :
% 3.05/1.62 ( Y
% 3.05/1.62 != ( product_Pair_a_a @ A4 @ B3 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_58_Pair__inject,axiom,
% 3.05/1.62 ! [A: a,B: a,A2: a,B2: a] :
% 3.05/1.62 ( ( ( product_Pair_a_a @ A @ B )
% 3.05/1.62 = ( product_Pair_a_a @ A2 @ B2 ) )
% 3.05/1.62 => ~ ( ( A = A2 )
% 3.05/1.62 => ( B != B2 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_59_prod__cases,axiom,
% 3.05/1.62 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 3.05/1.62 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 3.05/1.62 => ( P @ P2 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_60_surj__pair,axiom,
% 3.05/1.62 ! [P2: product_prod_a_a] :
% 3.05/1.62 ? [X3: a,Y4: a] :
% 3.05/1.62 ( P2
% 3.05/1.62 = ( product_Pair_a_a @ X3 @ Y4 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_61_f11,axiom,
% 3.05/1.62 ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ ( 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 @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_62_semiring_Osemiring__simprules_I11_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 3.05/1.62 = X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_63_semiring_Osemiring__simprules_I6_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 3.05/1.62 = X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_64_semiring_Ol__null,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 3.05/1.62 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_65_semiring_Or__null,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 3.05/1.62 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_66_semiring_Or__distr,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ Z @ ( add_a_b @ R3 @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_67_semiring_Ol__distr,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 3.05/1.62 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_68_add__pow__rdistr__int,axiom,
% 3.05/1.62 ! [A: a,B: a,K: int] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 3.05/1.62 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_69_add__pow__ldistr__int,axiom,
% 3.05/1.62 ! [A: a,B: a,K: int] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 3.05/1.62 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_70_inv__unique,axiom,
% 3.05/1.62 ! [Y: a,X: a,Y3: a] :
% 3.05/1.62 ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( Y = Y3 ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_71_one__unique,axiom,
% 3.05/1.62 ! [U: a] :
% 3.05/1.62 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
% 3.05/1.62 = X3 ) )
% 3.05/1.62 => ( U
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_72_add_Oint__pow__distrib,axiom,
% 3.05/1.62 ! [X: a,Y: a,I: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_73_add_Oint__pow__mult__distrib,axiom,
% 3.05/1.62 ! [X: a,Y: a,I: int] :
% 3.05/1.62 ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( add_a_b @ r @ Y @ X ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_74_semiring__simprules_I4_J,axiom,
% 3.05/1.62 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_75_add_Oint__pow__closed,axiom,
% 3.05/1.62 ! [X: a,I: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_76_add_Oint__pow__one,axiom,
% 3.05/1.62 ! [Z: int] :
% 3.05/1.62 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_77_l__one,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
% 3.05/1.62 = X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_78_r__one,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
% 3.05/1.62 = X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_79_semiring_Osemiring__simprules_I4_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 3.05/1.62 = X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_81_semiring_Osemiring__simprules_I3_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_82_semiring_Osemiring__simprules_I8_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_83_semiring_Osemiring__simprules_I1_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_84_semiring_Osemiring__simprules_I5_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 3.05/1.62 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_85_semiring_Osemiring__simprules_I7_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ X @ Y )
% 3.05/1.62 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_86_semiring_Osemiring__simprules_I12_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) )
% 3.05/1.62 = ( add_a_b @ R3 @ Y @ ( add_a_b @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_87_semiring_Osemiring__simprules_I2_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_88_cring__fieldI2,axiom,
% 3.05/1.62 ( ( ( zero_a_b @ r )
% 3.05/1.62 != ( one_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ! [A4: a] :
% 3.05/1.62 ( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( A4
% 3.05/1.62 != ( zero_a_b @ r ) )
% 3.05/1.62 => ? [X5: a] :
% 3.05/1.62 ( ( member_a @ X5 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ r @ A4 @ X5 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) ) )
% 3.05/1.62 => ( field_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_89_add_Oint__pow__mult,axiom,
% 3.05/1.62 ! [X: a,I: int,J: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_90_add__pow__ldistr,axiom,
% 3.05/1.62 ! [A: a,B: a,K: nat] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
% 3.05/1.62 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_91_add__pow__rdistr,axiom,
% 3.05/1.62 ! [A: a,B: a,K: nat] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
% 3.05/1.62 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_92_add_Ogroup__commutes__pow,axiom,
% 3.05/1.62 ! [X: a,Y: a,N: nat] :
% 3.05/1.62 ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( add_a_b @ r @ Y @ X ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
% 3.05/1.62 = ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_93_add_Opow__mult__distrib,axiom,
% 3.05/1.62 ! [X: a,Y: a,N: nat] :
% 3.05/1.62 ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( add_a_b @ r @ Y @ X ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_94_add_Onat__pow__distrib,axiom,
% 3.05/1.62 ! [X: a,Y: a,N: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_95_add_Onat__pow__comm,axiom,
% 3.05/1.62 ! [X: a,N: nat,M: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_96_add_Onat__pow__closed,axiom,
% 3.05/1.62 ! [X: a,N: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_97_add_Onat__pow__one,axiom,
% 3.05/1.62 ! [N: nat] :
% 3.05/1.62 ( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_98_one__not__zero,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( field_a_b @ R3 )
% 3.05/1.62 => ( ( one_a_ring_ext_a_b @ R3 )
% 3.05/1.62 != ( zero_a_b @ R3 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_99_integral,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,A: a,B: a] :
% 3.05/1.62 ( ( field_a_b @ R3 )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
% 3.05/1.62 = ( zero_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( A
% 3.05/1.62 = ( zero_a_b @ R3 ) )
% 3.05/1.62 | ( B
% 3.05/1.62 = ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_100_semiring_Oadd__pow__ldistr,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
% 3.05/1.62 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_101_semiring_Oadd__pow__rdistr,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 3.05/1.62 ( ( semiring_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
% 3.05/1.62 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_102_add_Onat__pow__mult,axiom,
% 3.05/1.62 ! [X: a,N: nat,M: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 3.05/1.62 = ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_103_add_Onat__pow__Suc2,axiom,
% 3.05/1.62 ! [X: a,N: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 3.05/1.62 = ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_104_add_Opow__eq__div2,axiom,
% 3.05/1.62 ! [X: a,M: nat,N: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_pow_a_b_nat @ r @ M @ X )
% 3.05/1.62 = ( add_pow_a_b_nat @ r @ N @ X ) )
% 3.05/1.62 => ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_105_minus__to__eq,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ r )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( a_minus_a_b @ r @ X @ Y )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( X = Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_106_ring_Oequality,axiom,
% 3.05/1.62 ! [R: partia1833973666xt_a_b,R2: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( ( partia1066395285xt_a_b @ R )
% 3.05/1.62 = ( partia1066395285xt_a_b @ R2 ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ R )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ R2 ) )
% 3.05/1.62 => ( ( ( one_a_ring_ext_a_b @ R )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ R2 ) )
% 3.05/1.62 => ( ( ( zero_a_b @ R )
% 3.05/1.62 = ( zero_a_b @ R2 ) )
% 3.05/1.62 => ( ( ( add_a_b @ R )
% 3.05/1.62 = ( add_a_b @ R2 ) )
% 3.05/1.62 => ( ( ( more_a_b @ R )
% 3.05/1.62 = ( more_a_b @ R2 ) )
% 3.05/1.62 => ( R = R2 ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_107_is__abelian__group,axiom,
% 3.05/1.62 abelian_group_a_b @ r ).
% 3.05/1.62
% 3.05/1.62 thf(fact_108_add_Onat__pow__Suc,axiom,
% 3.05/1.62 ! [N: nat,X: a] :
% 3.05/1.62 ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_109_abelian__groupE_I1_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_110_abelian__groupE_I3_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 3.05/1.62 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_111_abelian__groupE_I4_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ X @ Y )
% 3.05/1.62 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_112_abelian__groupE_I2_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_113_abelian__group_Ominus__closed,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_114_abelian__groupI,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b] :
% 3.05/1.62 ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( member_a @ ( add_a_b @ R3 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) )
% 3.05/1.62 => ( ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ! [Z2: a] :
% 3.05/1.62 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X3 @ Y4 ) @ Z2 )
% 3.05/1.62 = ( add_a_b @ R3 @ X3 @ ( add_a_b @ R3 @ Y4 @ Z2 ) ) ) ) ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ X3 @ Y4 )
% 3.05/1.62 = ( add_a_b @ R3 @ Y4 @ X3 ) ) ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X3 )
% 3.05/1.62 = X3 ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ? [Xa: a] :
% 3.05/1.62 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 & ( ( add_a_b @ R3 @ Xa @ X3 )
% 3.05/1.62 = ( zero_a_b @ R3 ) ) ) )
% 3.05/1.62 => ( abelian_group_a_b @ R3 ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_115_abelian__groupE_I5_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 3.05/1.62 = X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_116_abelian__groupE_I6_J,axiom,
% 3.05/1.62 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ R3 )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 3.05/1.62 & ( ( add_a_b @ R3 @ X3 @ X )
% 3.05/1.62 = ( zero_a_b @ R3 ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_117_abelian__group_Ofour__elem__comm,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a,D: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_minus_a_b @ G @ ( add_a_b @ G @ ( a_minus_a_b @ G @ A @ C ) @ B ) @ D )
% 3.05/1.62 = ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ ( add_a_b @ G @ A @ B ) @ C ) @ D ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_118_abelian__group_Oright__inv__add,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ C @ A ) @ B )
% 3.05/1.62 = ( a_minus_a_b @ G @ C @ ( add_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_119_abelian__group_Ominus__to__eq,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( ( a_minus_a_b @ G @ X @ Y )
% 3.05/1.62 = ( zero_a_b @ G ) )
% 3.05/1.62 => ( X = Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_120_diff__diff__left,axiom,
% 3.05/1.62 ! [I: nat,J: nat,K: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 3.05/1.62 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_121_Suc__diff__diff,axiom,
% 3.05/1.62 ! [M: nat,N: nat,K: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 3.05/1.62 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_122_diff__Suc__Suc,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.05/1.62 = ( minus_minus_nat @ M @ N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_123_add__Suc__right,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 3.05/1.62 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_124_nat_Oinject,axiom,
% 3.05/1.62 ! [X2: nat,Y2: nat] :
% 3.05/1.62 ( ( ( suc @ X2 )
% 3.05/1.62 = ( suc @ Y2 ) )
% 3.05/1.62 = ( X2 = Y2 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_125_old_Onat_Oinject,axiom,
% 3.05/1.62 ! [Nat: nat,Nat2: nat] :
% 3.05/1.62 ( ( ( suc @ Nat )
% 3.05/1.62 = ( suc @ Nat2 ) )
% 3.05/1.62 = ( Nat = Nat2 ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_126_Suc__inject,axiom,
% 3.05/1.62 ! [X: nat,Y: nat] :
% 3.05/1.62 ( ( ( suc @ X )
% 3.05/1.62 = ( suc @ Y ) )
% 3.05/1.62 => ( X = Y ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_127_n__not__Suc__n,axiom,
% 3.05/1.62 ! [N: nat] :
% 3.05/1.62 ( N
% 3.05/1.62 != ( suc @ N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_128_diff__commute,axiom,
% 3.05/1.62 ! [I: nat,J: nat,K: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 3.05/1.62 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_129_add__Suc,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 3.05/1.62 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_130_nat__arith_Osuc1,axiom,
% 3.05/1.62 ! [A3: nat,K: nat,A: nat] :
% 3.05/1.62 ( ( A3
% 3.05/1.62 = ( plus_plus_nat @ K @ A ) )
% 3.05/1.62 => ( ( suc @ A3 )
% 3.05/1.62 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_131_add__Suc__shift,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 3.05/1.62 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_132_zero__induct__lemma,axiom,
% 3.05/1.62 ! [P: nat > $o,K: nat,I: nat] :
% 3.05/1.62 ( ( P @ K )
% 3.05/1.62 => ( ! [N2: nat] :
% 3.05/1.62 ( ( P @ ( suc @ N2 ) )
% 3.05/1.62 => ( P @ N2 ) )
% 3.05/1.62 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_133_diff__add__inverse2,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 3.05/1.62 = M ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_134_diff__add__inverse,axiom,
% 3.05/1.62 ! [N: nat,M: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 3.05/1.62 = M ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_135_diff__cancel2,axiom,
% 3.05/1.62 ! [M: nat,K: nat,N: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 3.05/1.62 = ( minus_minus_nat @ M @ N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_136_Nat_Odiff__cancel,axiom,
% 3.05/1.62 ! [K: nat,M: nat,N: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.05/1.62 = ( minus_minus_nat @ M @ N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_137_add__diff__cancel__right_H,axiom,
% 3.05/1.62 ! [A: nat,B: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 3.05/1.62 = A ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_138_add__diff__cancel__right_H,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 3.05/1.62 = A ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_139_add__diff__cancel__right,axiom,
% 3.05/1.62 ! [A: nat,C: nat,B: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 3.05/1.62 = ( minus_minus_nat @ A @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_140_add__diff__cancel__right,axiom,
% 3.05/1.62 ! [A: int,C: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 3.05/1.62 = ( minus_minus_int @ A @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_141_add__diff__cancel__left_H,axiom,
% 3.05/1.62 ! [A: nat,B: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 3.05/1.62 = B ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_142_add__diff__cancel__left_H,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 3.05/1.62 = B ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_143_add__diff__cancel__left,axiom,
% 3.05/1.62 ! [C: nat,A: nat,B: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 3.05/1.62 = ( minus_minus_nat @ A @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_144_add__diff__cancel__left,axiom,
% 3.05/1.62 ! [C: int,A: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 3.05/1.62 = ( minus_minus_int @ A @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_145_add__right__cancel,axiom,
% 3.05/1.62 ! [B: int,A: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ B @ A )
% 3.05/1.62 = ( plus_plus_int @ C @ A ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_146_add__right__cancel,axiom,
% 3.05/1.62 ! [B: nat,A: nat,C: nat] :
% 3.05/1.62 ( ( ( plus_plus_nat @ B @ A )
% 3.05/1.62 = ( plus_plus_nat @ C @ A ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_147_add__left__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ A @ B )
% 3.05/1.62 = ( plus_plus_int @ A @ C ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_148_add__left__cancel,axiom,
% 3.05/1.62 ! [A: nat,B: nat,C: nat] :
% 3.05/1.62 ( ( ( plus_plus_nat @ A @ B )
% 3.05/1.62 = ( plus_plus_nat @ A @ C ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_149_add__diff__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 3.05/1.62 = A ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_150_diff__add__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 3.05/1.62 = A ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_151_add__right__imp__eq,axiom,
% 3.05/1.62 ! [B: int,A: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ B @ A )
% 3.05/1.62 = ( plus_plus_int @ C @ A ) )
% 3.05/1.62 => ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_152_add__right__imp__eq,axiom,
% 3.05/1.62 ! [B: nat,A: nat,C: nat] :
% 3.05/1.62 ( ( ( plus_plus_nat @ B @ A )
% 3.05/1.62 = ( plus_plus_nat @ C @ A ) )
% 3.05/1.62 => ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_153_add__left__imp__eq,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ A @ B )
% 3.05/1.62 = ( plus_plus_int @ A @ C ) )
% 3.05/1.62 => ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_154_add__left__imp__eq,axiom,
% 3.05/1.62 ! [A: nat,B: nat,C: nat] :
% 3.05/1.62 ( ( ( plus_plus_nat @ A @ B )
% 3.05/1.62 = ( plus_plus_nat @ A @ C ) )
% 3.05/1.62 => ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_155_add_Oleft__commute,axiom,
% 3.05/1.62 ! [B: int,A: int,C: int] :
% 3.05/1.62 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 3.05/1.62 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_156_add_Oleft__commute,axiom,
% 3.05/1.62 ! [B: nat,A: nat,C: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 3.05/1.62 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_157_add_Ocommute,axiom,
% 3.05/1.62 ( plus_plus_int
% 3.05/1.62 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_158_add_Ocommute,axiom,
% 3.05/1.62 ( plus_plus_nat
% 3.05/1.62 = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_159_group__add__class_Oadd_Oright__cancel,axiom,
% 3.05/1.62 ! [B: int,A: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ B @ A )
% 3.05/1.62 = ( plus_plus_int @ C @ A ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_160_add_Oleft__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ A @ B )
% 3.05/1.62 = ( plus_plus_int @ A @ C ) )
% 3.05/1.62 = ( B = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_161_add_Oassoc,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 3.05/1.62 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_162_add_Oassoc,axiom,
% 3.05/1.62 ! [A: nat,B: nat,C: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 3.05/1.62 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_163_group__cancel_Oadd2,axiom,
% 3.05/1.62 ! [B5: int,K: int,B: int,A: int] :
% 3.05/1.62 ( ( B5
% 3.05/1.62 = ( plus_plus_int @ K @ B ) )
% 3.05/1.62 => ( ( plus_plus_int @ A @ B5 )
% 3.05/1.62 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_164_group__cancel_Oadd2,axiom,
% 3.05/1.62 ! [B5: nat,K: nat,B: nat,A: nat] :
% 3.05/1.62 ( ( B5
% 3.05/1.62 = ( plus_plus_nat @ K @ B ) )
% 3.05/1.62 => ( ( plus_plus_nat @ A @ B5 )
% 3.05/1.62 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_165_group__cancel_Oadd1,axiom,
% 3.05/1.62 ! [A3: int,K: int,A: int,B: int] :
% 3.05/1.62 ( ( A3
% 3.05/1.62 = ( plus_plus_int @ K @ A ) )
% 3.05/1.62 => ( ( plus_plus_int @ A3 @ B )
% 3.05/1.62 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_166_group__cancel_Oadd1,axiom,
% 3.05/1.62 ! [A3: nat,K: nat,A: nat,B: nat] :
% 3.05/1.62 ( ( A3
% 3.05/1.62 = ( plus_plus_nat @ K @ A ) )
% 3.05/1.62 => ( ( plus_plus_nat @ A3 @ B )
% 3.05/1.62 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_167_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.05/1.62 ! [I: int,J: int,K: int,L: int] :
% 3.05/1.62 ( ( ( I = J )
% 3.05/1.62 & ( K = L ) )
% 3.05/1.62 => ( ( plus_plus_int @ I @ K )
% 3.05/1.62 = ( plus_plus_int @ J @ L ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_168_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.05/1.62 ! [I: nat,J: nat,K: nat,L: nat] :
% 3.05/1.62 ( ( ( I = J )
% 3.05/1.62 & ( K = L ) )
% 3.05/1.62 => ( ( plus_plus_nat @ I @ K )
% 3.05/1.62 = ( plus_plus_nat @ J @ L ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 3.05/1.62 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_170_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.05/1.62 ! [A: nat,B: nat,C: nat] :
% 3.05/1.62 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 3.05/1.62 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_171_diff__right__commute,axiom,
% 3.05/1.62 ! [A: nat,C: nat,B: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 3.05/1.62 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_172_diff__right__commute,axiom,
% 3.05/1.62 ! [A: int,C: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 3.05/1.62 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_173_diff__eq__diff__eq,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int,D: int] :
% 3.05/1.62 ( ( ( minus_minus_int @ A @ B )
% 3.05/1.62 = ( minus_minus_int @ C @ D ) )
% 3.05/1.62 => ( ( A = B )
% 3.05/1.62 = ( C = D ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_174_group__cancel_Osub1,axiom,
% 3.05/1.62 ! [A3: int,K: int,A: int,B: int] :
% 3.05/1.62 ( ( A3
% 3.05/1.62 = ( plus_plus_int @ K @ A ) )
% 3.05/1.62 => ( ( minus_minus_int @ A3 @ B )
% 3.05/1.62 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_175_diff__eq__eq,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( ( minus_minus_int @ A @ B )
% 3.05/1.62 = C )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( plus_plus_int @ C @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_176_eq__diff__eq,axiom,
% 3.05/1.62 ! [A: int,C: int,B: int] :
% 3.05/1.62 ( ( A
% 3.05/1.62 = ( minus_minus_int @ C @ B ) )
% 3.05/1.62 = ( ( plus_plus_int @ A @ B )
% 3.05/1.62 = C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_177_add__diff__eq,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 3.05/1.62 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_178_diff__diff__eq2,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 3.05/1.62 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_179_diff__add__eq,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 3.05/1.62 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_180_diff__add__eq__diff__diff__swap,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 3.05/1.62 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_181_diff__diff__add,axiom,
% 3.05/1.62 ! [A: nat,B: nat,C: nat] :
% 3.05/1.62 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 3.05/1.62 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_182_diff__diff__add,axiom,
% 3.05/1.62 ! [A: int,B: int,C: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 3.05/1.62 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_183_add__implies__diff,axiom,
% 3.05/1.62 ! [C: nat,B: nat,A: nat] :
% 3.05/1.62 ( ( ( plus_plus_nat @ C @ B )
% 3.05/1.62 = A )
% 3.05/1.62 => ( C
% 3.05/1.62 = ( minus_minus_nat @ A @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_184_add__implies__diff,axiom,
% 3.05/1.62 ! [C: int,B: int,A: int] :
% 3.05/1.62 ( ( ( plus_plus_int @ C @ B )
% 3.05/1.62 = A )
% 3.05/1.62 => ( C
% 3.05/1.62 = ( minus_minus_int @ A @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_185_Units__l__inv__ex,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_186_Units__r__inv__ex,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_187_group__l__invI,axiom,
% 3.05/1.62 ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ? [Xa: a] :
% 3.05/1.62 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ r @ Xa @ X3 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) )
% 3.05/1.62 => ( group_a_ring_ext_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_188_Units__closed,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_189_Units__inv__comm,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_190_Units__m__closed,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_191_Units__one__closed,axiom,
% 3.05/1.62 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_192_Units__l__cancel,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
% 3.05/1.62 = ( Y = Z ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_193_groupI,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit] :
% 3.05/1.62 ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ ( partia1955795460t_unit @ G ) ) ) )
% 3.05/1.62 => ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ! [Z2: a] :
% 3.05/1.62 ( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ Z2 )
% 3.05/1.62 = ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y4 @ Z2 ) ) ) ) ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X3 )
% 3.05/1.62 = X3 ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ? [Xa: a] :
% 3.05/1.62 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 & ( ( mult_a_Product_unit @ G @ Xa @ X3 )
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) )
% 3.05/1.62 => ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_194_groupI,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b] :
% 3.05/1.62 ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 3.05/1.62 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ! [Y4: a] :
% 3.05/1.62 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ! [Z2: a] :
% 3.05/1.62 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ Z2 )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y4 @ Z2 ) ) ) ) ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X3 )
% 3.05/1.62 = X3 ) )
% 3.05/1.62 => ( ! [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ? [Xa: a] :
% 3.05/1.62 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X3 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 3.05/1.62 => ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_195_group_Oinv__comm,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,Y: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( ( mult_a_Product_unit @ G @ X @ Y )
% 3.05/1.62 = ( one_a_Product_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( mult_a_Product_unit @ G @ Y @ X )
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_196_group_Oinv__comm,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_197_group_Ol__inv__ex,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 & ( ( mult_a_Product_unit @ G @ X3 @ X )
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_198_group_Ol__inv__ex,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ G @ X3 @ X )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_199_Group_Ogroup_Oright__cancel,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( ( mult_a_Product_unit @ G @ Y @ X )
% 3.05/1.62 = ( mult_a_Product_unit @ G @ Z @ X ) )
% 3.05/1.62 = ( Y = Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_200_Group_Ogroup_Oright__cancel,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
% 3.05/1.62 = ( Y = Z ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_201_group_OUnits__eq,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( units_a_Product_unit @ G )
% 3.05/1.62 = ( partia1955795460t_unit @ G ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_202_group_OUnits__eq,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( units_a_ring_ext_a_b @ G )
% 3.05/1.62 = ( partia1066395285xt_a_b @ G ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_203_group_Or__cancel__one_H,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( mult_a_Product_unit @ G @ A @ X ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_204_group_Or__cancel__one_H,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ A @ X ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_205_group_Ol__cancel__one_H,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( mult_a_Product_unit @ G @ X @ A ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_206_group_Ol__cancel__one_H,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( X
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ X @ A ) )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_207_group_Or__cancel__one,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( ( mult_a_Product_unit @ G @ A @ X )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_208_group_Or__cancel__one,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_209_group_Ol__cancel__one,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( ( mult_a_Product_unit @ G @ X @ A )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_210_group_Ol__cancel__one,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
% 3.05/1.62 = X )
% 3.05/1.62 = ( A
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_211_group_Or__inv__ex,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a] :
% 3.05/1.62 ( ( group_a_Product_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 & ( ( mult_a_Product_unit @ G @ X @ X3 )
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_212_group_Or__inv__ex,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( group_a_ring_ext_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ? [X3: a] :
% 3.05/1.62 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 & ( ( mult_a_ring_ext_a_b @ G @ X @ X3 )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_213_units__group,axiom,
% 3.05/1.62 group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_214_inv__add,axiom,
% 3.05/1.62 ! [A: a,B: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 3.05/1.62 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_215_r__neg2,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 3.05/1.62 = Y ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_216_r__neg1,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = Y ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_217_local_Ominus__add,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_218_add_Oinv__solve__right_H,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 3.05/1.62 = A )
% 3.05/1.62 = ( B
% 3.05/1.62 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_219_add_Oinv__solve__right,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( A
% 3.05/1.62 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 3.05/1.62 = ( B
% 3.05/1.62 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_220_add_Oinv__solve__left_H,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 3.05/1.62 = A )
% 3.05/1.62 = ( C
% 3.05/1.62 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_221_add_Oinv__solve__left,axiom,
% 3.05/1.62 ! [A: a,B: a,C: a] :
% 3.05/1.62 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( A
% 3.05/1.62 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 3.05/1.62 = ( C
% 3.05/1.62 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_222_add_Oinv__mult__group,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_223_a__transpose__inv,axiom,
% 3.05/1.62 ! [X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = Z )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 3.05/1.62 = Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_224_r__minus,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_225_l__minus,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_226_add_Onat__pow__inv,axiom,
% 3.05/1.62 ! [X: a,I: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_227_add_Oint__pow__inv,axiom,
% 3.05/1.62 ! [X: a,I: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_228_minus__eq,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( a_minus_a_b @ r @ X @ Y )
% 3.05/1.62 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_229_sum__zero__eq__neg,axiom,
% 3.05/1.62 ! [X: a,Y: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( add_a_b @ r @ X @ Y )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( X
% 3.05/1.62 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_230_r__neg,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_231_minus__equality,axiom,
% 3.05/1.62 ! [Y: a,X: a] :
% 3.05/1.62 ( ( ( add_a_b @ r @ Y @ X )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_inv_a_b @ r @ X )
% 3.05/1.62 = Y ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_232_l__neg,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_233_add_Oint__pow__diff,axiom,
% 3.05/1.62 ! [X: a,N: int,M: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
% 3.05/1.62 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_234_local_Ominus__minus,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 3.05/1.62 = X ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_235_add_Oinv__closed,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_236_local_Ominus__zero,axiom,
% 3.05/1.62 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 3.05/1.62 = ( zero_a_b @ r ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_237_add_Oinv__eq__1__iff,axiom,
% 3.05/1.62 ! [X: a] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( ( a_inv_a_b @ r @ X )
% 3.05/1.62 = ( zero_a_b @ r ) )
% 3.05/1.62 = ( X
% 3.05/1.62 = ( zero_a_b @ r ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_238_Units__minus__one__closed,axiom,
% 3.05/1.62 member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_239_units__of__mult,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit] :
% 3.05/1.62 ( ( mult_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 3.05/1.62 = ( mult_a_Product_unit @ G ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_240_units__of__mult,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_241_abelian__group_Oa__inv__closed,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_242_abelian__group_Ominus__minus,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
% 3.05/1.62 = X ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_243_a__minus__def,axiom,
% 3.05/1.62 ( a_minus_a_b
% 3.05/1.62 = ( ^ [R4: partia1833973666xt_a_b,X4: a,Y5: a] : ( add_a_b @ R4 @ X4 @ ( a_inv_a_b @ R4 @ Y5 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_244_units__of__one,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit] :
% 3.05/1.62 ( ( one_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 3.05/1.62 = ( one_a_Product_unit @ G ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_245_units__of__one,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.05/1.62 = ( one_a_ring_ext_a_b @ G ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_246_abelian__group_Ominus__add,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
% 3.05/1.62 = ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_247_abelian__group_Or__neg2,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
% 3.05/1.62 = Y ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_248_abelian__group_Or__neg1,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
% 3.05/1.62 = Y ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_249_abelian__group_Ominus__eq,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( a_minus_a_b @ G @ X @ Y )
% 3.05/1.62 = ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_250_abelian__group_Ol__neg,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
% 3.05/1.62 = ( zero_a_b @ G ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_251_abelian__group_Or__neg,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
% 3.05/1.62 = ( zero_a_b @ G ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_252_abelian__group_Ominus__equality,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,Y: a,X: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( ( add_a_b @ G @ Y @ X )
% 3.05/1.62 = ( zero_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_inv_a_b @ G @ X )
% 3.05/1.62 = Y ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_253_abelian__group_Oinv__add,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,A: a,B: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ A @ B ) )
% 3.05/1.62 = ( a_minus_a_b @ G @ ( a_inv_a_b @ G @ A ) @ B ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_254_abelian__group_Oa__transpose__inv,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( abelian_group_a_b @ G )
% 3.05/1.62 => ( ( ( add_a_b @ G @ X @ Y )
% 3.05/1.62 = Z )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Z )
% 3.05/1.62 = Y ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_255_add_Oint__pow__neg,axiom,
% 3.05/1.62 ! [X: a,I: int] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_256_add__minus__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 3.05/1.62 = B ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_257_minus__add__cancel,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 3.05/1.62 = B ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_258_minus__add__distrib,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 3.05/1.62 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_259_minus__diff__eq,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 3.05/1.62 = ( minus_minus_int @ B @ A ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_260_diff__minus__eq__add,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 3.05/1.62 = ( plus_plus_int @ A @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_261_uminus__add__conv__diff,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 3.05/1.62 = ( minus_minus_int @ B @ A ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_262_group__cancel_Osub2,axiom,
% 3.05/1.62 ! [B5: int,K: int,B: int,A: int] :
% 3.05/1.62 ( ( B5
% 3.05/1.62 = ( plus_plus_int @ K @ B ) )
% 3.05/1.62 => ( ( minus_minus_int @ A @ B5 )
% 3.05/1.62 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_263_diff__conv__add__uminus,axiom,
% 3.05/1.62 ( minus_minus_int
% 3.05/1.62 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_264_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 3.05/1.62 ( minus_minus_int
% 3.05/1.62 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_265_group__cancel_Oneg1,axiom,
% 3.05/1.62 ! [A3: int,K: int,A: int] :
% 3.05/1.62 ( ( A3
% 3.05/1.62 = ( plus_plus_int @ K @ A ) )
% 3.05/1.62 => ( ( uminus_uminus_int @ A3 )
% 3.05/1.62 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_266_add_Oinverse__distrib__swap,axiom,
% 3.05/1.62 ! [A: int,B: int] :
% 3.05/1.62 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 3.05/1.62 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_267_minus__diff__commute,axiom,
% 3.05/1.62 ! [B: int,A: int] :
% 3.05/1.62 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 3.05/1.62 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_268_add_Oint__pow__neg__int,axiom,
% 3.05/1.62 ! [X: a,N: nat] :
% 3.05/1.62 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.05/1.62 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) @ X )
% 3.05/1.62 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_269_units__comm__group,axiom,
% 3.05/1.62 comm_g1684316527t_unit @ ( units_1411277569xt_a_b @ r ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_270_of__nat__eq__iff,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( ( semiri1382578993at_nat @ M )
% 3.05/1.62 = ( semiri1382578993at_nat @ N ) )
% 3.05/1.62 = ( M = N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_271_of__nat__eq__iff,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( ( semiri2019852685at_int @ M )
% 3.05/1.62 = ( semiri2019852685at_int @ N ) )
% 3.05/1.62 = ( M = N ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_272_of__nat__add,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( semiri1382578993at_nat @ ( plus_plus_nat @ M @ N ) )
% 3.05/1.62 = ( plus_plus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_273_of__nat__add,axiom,
% 3.05/1.62 ! [M: nat,N: nat] :
% 3.05/1.62 ( ( semiri2019852685at_int @ ( plus_plus_nat @ M @ N ) )
% 3.05/1.62 = ( plus_plus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_274_comm__groupE_I4_J,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,Y: a] :
% 3.05/1.62 ( ( comm_g1684316527t_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( mult_a_Product_unit @ G @ X @ Y )
% 3.05/1.62 = ( mult_a_Product_unit @ G @ Y @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_275_comm__groupE_I4_J,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( comm_g791708116xt_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ Y @ X ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_276_comm__groupE_I3_J,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( comm_g1684316527t_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ Z )
% 3.05/1.62 = ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_277_comm__groupE_I3_J,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.05/1.62 ( ( comm_g791708116xt_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
% 3.05/1.62 = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_278_comm__groupE_I1_J,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit,X: a,Y: a] :
% 3.05/1.62 ( ( comm_g1684316527t_unit @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.05/1.62 => ( member_a @ ( mult_a_Product_unit @ G @ X @ Y ) @ ( partia1955795460t_unit @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_279_comm__groupE_I1_J,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.05/1.62 ( ( comm_g791708116xt_a_b @ G )
% 3.05/1.62 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.05/1.62 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_280_comm__groupE_I2_J,axiom,
% 3.05/1.62 ! [G: partia96731725t_unit] :
% 3.05/1.62 ( ( comm_g1684316527t_unit @ G )
% 3.05/1.62 => ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(fact_281_comm__groupE_I2_J,axiom,
% 3.05/1.62 ! [G: partia1833973666xt_a_b] :
% 3.05/1.62 ( ( comm_g791708116xt_a_b @ G )
% 3.05/1.62 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 3.05/1.62
% 3.05/1.62 thf(conj_0,conjecture,
% 3.05/1.62 member449909584od_a_a @ ( product_Pair_a_a @ ( 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 @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ).
% 3.05/1.62 ------------------------
% 3.05/1.64 % (4960)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/2Mi)
% 3.05/1.64 % (4959)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/2Mi)
% 3.05/1.65 % (4959)Instruction limit reached!
% 3.05/1.65 % (4959)------------------------------
% 3.05/1.65 % (4959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (4959)Termination reason: Unknown
% 3.05/1.65 % (4960)Instruction limit reached!
% 3.05/1.65 % (4960)------------------------------
% 3.05/1.65 % (4960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (4960)Termination reason: Unknown
% 3.05/1.65 % (4960)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (4960)Memory used [KB]: 1407
% 3.05/1.65 % (4960)Time elapsed: 0.003 s
% 3.05/1.65 % (4960)Instructions burned: 4 (million)
% 3.05/1.65 % (4960)------------------------------
% 3.05/1.65 % (4960)------------------------------
% 3.05/1.65 % (4959)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (4959)Memory used [KB]: 1279
% 3.05/1.65 % (4959)Time elapsed: 0.003 s
% 3.05/1.65 % (4959)Instructions burned: 3 (million)
% 3.05/1.65 % (4959)------------------------------
% 3.05/1.65 % (4959)------------------------------
% 3.05/1.65 % (4958)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/27Mi)
% 3.05/1.65 % (4962)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/18Mi)
% 3.05/1.65 % (4961)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/275Mi)
% 3.05/1.65 % (4956)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/183Mi)
% 3.05/1.65 % (4957)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/4Mi)
% 3.05/1.65 % (4957)Instruction limit reached!
% 3.05/1.65 % (4957)------------------------------
% 3.05/1.65 % (4957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (4957)Termination reason: Unknown
% 3.05/1.65 % (4957)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (4957)Memory used [KB]: 1407
% 3.05/1.65 % (4957)Time elapsed: 0.004 s
% 3.05/1.65 % (4957)Instructions burned: 4 (million)
% 3.05/1.65 % (4957)------------------------------
% 3.05/1.65 % (4957)------------------------------
% 3.05/1.66 % (4966)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/3Mi)
% 3.05/1.66 % (4966)Instruction limit reached!
% 3.05/1.66 % (4966)------------------------------
% 3.05/1.66 % (4966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (4966)Termination reason: Unknown
% 3.05/1.66 % (4966)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (4966)Memory used [KB]: 1279
% 3.05/1.66 % (4966)Time elapsed: 0.002 s
% 3.05/1.66 % (4966)Instructions burned: 3 (million)
% 3.05/1.66 % (4966)------------------------------
% 3.05/1.66 % (4966)------------------------------
% 3.05/1.66 % (4962)Instruction limit reached!
% 3.05/1.66 % (4962)------------------------------
% 3.05/1.66 % (4962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (4962)Termination reason: Unknown
% 3.05/1.66 % (4962)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (4962)Memory used [KB]: 1535
% 3.05/1.66 % (4962)Time elapsed: 0.011 s
% 3.05/1.66 % (4962)Instructions burned: 19 (million)
% 3.05/1.66 % (4962)------------------------------
% 3.05/1.66 % (4962)------------------------------
% 3.05/1.66 % (4967)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/37Mi)
% 3.05/1.66 % (4958)Instruction limit reached!
% 3.05/1.66 % (4958)------------------------------
% 3.05/1.66 % (4958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (4958)Termination reason: Unknown
% 3.05/1.66 % (4958)Termination phase: Property scanning
% 3.05/1.66
% 3.05/1.66 % (4958)Memory used [KB]: 1535
% 3.05/1.66 % (4958)Time elapsed: 0.014 s
% 3.05/1.66 % (4958)Instructions burned: 28 (million)
% 3.05/1.66 % (4958)------------------------------
% 3.05/1.66 % (4958)------------------------------
% 3.05/1.67 % (4969)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/15Mi)
% 3.05/1.67 % (4972)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/3Mi)
% 3.05/1.67 % (4967)Instruction limit reached!
% 3.05/1.67 % (4967)------------------------------
% 3.05/1.67 % (4967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (4967)Termination reason: Unknown
% 3.05/1.67 % (4967)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (4967)Memory used [KB]: 1918
% 3.05/1.67 % (4967)Time elapsed: 0.012 s
% 3.05/1.67 % (4967)Instructions burned: 40 (million)
% 3.05/1.67 % (4967)------------------------------
% 3.05/1.67 % (4967)------------------------------
% 3.05/1.67 % (4972)Instruction limit reached!
% 3.05/1.67 % (4972)------------------------------
% 3.05/1.67 % (4972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (4972)Termination reason: Unknown
% 3.05/1.67 % (4972)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (4972)Memory used [KB]: 1279
% 3.05/1.67 % (4972)Time elapsed: 0.004 s
% 3.05/1.67 % (4972)Instructions burned: 4 (million)
% 3.05/1.67 % (4972)------------------------------
% 3.05/1.67 % (4972)------------------------------
% 3.05/1.68 % (4973)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/1041Mi)
% 3.05/1.68 % (4975)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/7Mi)
% 3.05/1.68 % (4969)Instruction limit reached!
% 3.05/1.68 % (4969)------------------------------
% 3.05/1.68 % (4969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (4969)Termination reason: Unknown
% 3.05/1.68 % (4969)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (4969)Memory used [KB]: 1535
% 3.05/1.68 % (4969)Time elapsed: 0.010 s
% 3.05/1.68 % (4969)Instructions burned: 16 (million)
% 3.05/1.68 % (4969)------------------------------
% 3.05/1.68 % (4969)------------------------------
% 3.05/1.68 % (4975)Instruction limit reached!
% 3.05/1.68 % (4975)------------------------------
% 3.05/1.68 % (4975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (4975)Termination reason: Unknown
% 3.05/1.68 % (4975)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (4975)Memory used [KB]: 1407
% 3.05/1.68 % (4975)Time elapsed: 0.005 s
% 3.05/1.68 % (4975)Instructions burned: 7 (million)
% 3.05/1.68 % (4975)------------------------------
% 3.05/1.68 % (4975)------------------------------
% 3.05/1.68 % (4977)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/16Mi)
% 3.05/1.69 % (4978)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/3Mi)
% 3.05/1.69 % (4961)Refutation not found, incomplete strategy
% 3.05/1.69 % (4961)------------------------------
% 3.05/1.69 % (4961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (4961)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.69
% 3.05/1.69
% 3.05/1.69 % (4961)Memory used [KB]: 6524
% 3.05/1.69 % (4961)Time elapsed: 0.041 s
% 3.05/1.69 % (4961)Instructions burned: 99 (million)
% 3.05/1.69 % (4961)------------------------------
% 3.05/1.69 % (4961)------------------------------
% 3.05/1.69 % (4977)Instruction limit reached!
% 3.05/1.69 % (4977)------------------------------
% 3.05/1.69 % (4977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (4977)Termination reason: Unknown
% 3.05/1.69 % (4977)Termination phase: shuffling
% 3.05/1.69
% 3.05/1.69 % (4977)Memory used [KB]: 1535
% 3.05/1.69 % (4977)Time elapsed: 0.010 s
% 3.05/1.69 % (4977)Instructions burned: 16 (million)
% 3.05/1.69 % (4977)------------------------------
% 3.05/1.69 % (4977)------------------------------
% 3.05/1.69 % (4978)Instruction limit reached!
% 3.05/1.69 % (4978)------------------------------
% 3.05/1.69 % (4978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (4978)Termination reason: Unknown
% 3.05/1.69 % (4978)Termination phase: shuffling
% 3.05/1.69
% 3.05/1.69 % (4978)Memory used [KB]: 1279
% 3.05/1.69 % (4978)Time elapsed: 0.003 s
% 3.05/1.69 % (4978)Instructions burned: 3 (million)
% 3.05/1.69 % (4978)------------------------------
% 3.05/1.69 % (4978)------------------------------
% 3.05/1.69 % (4982)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/3Mi)
% 3.05/1.70 % (4982)Instruction limit reached!
% 3.05/1.70 % (4982)------------------------------
% 3.05/1.70 % (4982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.70 % (4982)Termination reason: Unknown
% 3.05/1.70 % (4982)Termination phase: shuffling
% 3.05/1.70
% 3.05/1.70 % (4982)Memory used [KB]: 1279
% 3.05/1.70 % (4982)Time elapsed: 0.003 s
% 3.05/1.70 % (4982)Instructions burned: 3 (million)
% 3.05/1.70 % (4982)------------------------------
% 3.05/1.70 % (4982)------------------------------
% 3.07/1.70 % (4984)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/7Mi)
% 3.07/1.70 % (4985)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/3Mi)
% 3.07/1.70 % (4984)Instruction limit reached!
% 3.07/1.70 % (4984)------------------------------
% 3.07/1.70 % (4984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.70 % (4984)Termination reason: Unknown
% 3.07/1.70 % (4984)Termination phase: shuffling
% 3.07/1.70
% 3.07/1.70 % (4984)Memory used [KB]: 1407
% 3.07/1.70 % (4984)Time elapsed: 0.005 s
% 3.07/1.70 % (4984)Instructions burned: 9 (million)
% 3.07/1.70 % (4984)------------------------------
% 3.07/1.70 % (4984)------------------------------
% 3.07/1.70 % (4985)Instruction limit reached!
% 3.07/1.70 % (4985)------------------------------
% 3.07/1.70 % (4985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.70 % (4985)Termination reason: Unknown
% 3.07/1.70 % (4985)Termination phase: shuffling
% 3.07/1.70
% 3.07/1.70 % (4985)Memory used [KB]: 1279
% 3.07/1.70 % (4985)Time elapsed: 0.003 s
% 3.07/1.70 % (4985)Instructions burned: 3 (million)
% 3.07/1.70 % (4985)------------------------------
% 3.07/1.70 % (4985)------------------------------
% 3.07/1.70 % (4986)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/4Mi)
% 3.07/1.70 % (4986)Instruction limit reached!
% 3.07/1.70 % (4986)------------------------------
% 3.07/1.70 % (4986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.70 % (4986)Termination reason: Unknown
% 3.07/1.70 % (4986)Termination phase: shuffling
% 3.07/1.70
% 3.07/1.70 % (4986)Memory used [KB]: 1279
% 3.07/1.70 % (4986)Time elapsed: 0.003 s
% 3.07/1.70 % (4986)Instructions burned: 4 (million)
% 3.07/1.70 % (4986)------------------------------
% 3.07/1.70 % (4986)------------------------------
% 3.07/1.71 % (4987)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/18Mi)
% 3.07/1.71 % (4989)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/710Mi)
% 3.07/1.71 % (4994)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/21Mi)
% 3.07/1.71 % (4987)Instruction limit reached!
% 3.07/1.71 % (4987)------------------------------
% 3.07/1.71 % (4987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.71 % (4987)Termination reason: Unknown
% 3.07/1.71 % (4987)Termination phase: shuffling
% 3.07/1.71
% 3.07/1.71 % (4987)Memory used [KB]: 1535
% 3.07/1.71 % (4987)Time elapsed: 0.010 s
% 3.07/1.71 % (4987)Instructions burned: 18 (million)
% 3.07/1.71 % (4987)------------------------------
% 3.07/1.71 % (4987)------------------------------
% 3.07/1.72 % (4992)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/6Mi)
% 3.07/1.72 % (4993)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/902Mi)
% 3.07/1.72 % (4992)Instruction limit reached!
% 3.07/1.72 % (4992)------------------------------
% 3.07/1.72 % (4992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.72 % (4994)Instruction limit reached!
% 3.07/1.72 % (4994)------------------------------
% 3.07/1.72 % (4994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.72 % (4992)Termination reason: Unknown
% 3.07/1.72 % (4992)Termination phase: shuffling
% 3.07/1.72
% 3.07/1.72 % (4992)Memory used [KB]: 1407
% 3.07/1.72 % (4992)Time elapsed: 0.005 s
% 3.07/1.72 % (4992)Instructions burned: 6 (million)
% 3.07/1.72 % (4992)------------------------------
% 3.07/1.72 % (4992)------------------------------
% 3.07/1.72 % (4994)Termination reason: Unknown
% 3.07/1.72 % (4994)Termination phase: shuffling
% 3.07/1.72
% 3.07/1.72 % (4994)Memory used [KB]: 1535
% 3.07/1.72 % (4994)Time elapsed: 0.008 s
% 3.07/1.72 % (4994)Instructions burned: 22 (million)
% 3.07/1.72 % (4994)------------------------------
% 3.07/1.72 % (4994)------------------------------
% 3.07/1.73 % (4999)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/6Mi)
% 3.07/1.73 % (4997)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/5Mi)
% 3.07/1.73 % (4999)Instruction limit reached!
% 3.07/1.73 % (4999)------------------------------
% 3.07/1.73 % (4999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.73 % (4999)Termination reason: Unknown
% 3.07/1.73 % (4999)Termination phase: shuffling
% 3.07/1.73
% 3.07/1.73 % (4999)Memory used [KB]: 1407
% 3.07/1.73 % (4999)Time elapsed: 0.003 s
% 3.07/1.73 % (4999)Instructions burned: 7 (million)
% 3.07/1.73 % (4999)------------------------------
% 3.07/1.73 % (4999)------------------------------
% 3.07/1.73 % (4997)Instruction limit reached!
% 3.07/1.73 % (4997)------------------------------
% 3.07/1.73 % (4997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.73 % (4997)Termination reason: Unknown
% 3.07/1.73 % (4997)Termination phase: shuffling
% 3.07/1.73
% 3.07/1.73 % (4997)Memory used [KB]: 1407
% 3.07/1.73 % (4997)Time elapsed: 0.004 s
% 3.07/1.73 % (4997)Instructions burned: 6 (million)
% 3.07/1.73 % (4997)------------------------------
% 3.07/1.73 % (4997)------------------------------
% 3.07/1.74 % (5000)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/377Mi)
% 3.07/1.74 % (4956)Instruction limit reached!
% 3.07/1.74 % (4956)------------------------------
% 3.07/1.74 % (4956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.74 % (4956)Termination reason: Unknown
% 3.07/1.74 % (4956)Termination phase: Saturation
% 3.07/1.74
% 3.07/1.74 % (4956)Memory used [KB]: 6780
% 3.07/1.74 % (4956)Time elapsed: 0.088 s
% 3.07/1.74 % (4956)Instructions burned: 183 (million)
% 3.07/1.74 % (4956)------------------------------
% 3.07/1.74 % (4956)------------------------------
% 3.07/1.74 % (5001)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4744 for (2999ds/779Mi)
% 3.07/1.75 % (5004)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/19Mi)
% 3.07/1.75 % (5005)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/879Mi)
% 3.07/1.76 % (5004)Instruction limit reached!
% 3.07/1.76 % (5004)------------------------------
% 3.07/1.76 % (5004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.76 % (5004)Termination reason: Unknown
% 3.07/1.76 % (5004)Termination phase: shuffling
% 3.07/1.76
% 3.07/1.76 % (5004)Memory used [KB]: 1663
% 3.07/1.76 % (5004)Time elapsed: 0.011 s
% 3.07/1.76 % (5004)Instructions burned: 21 (million)
% 3.07/1.76 % (5004)------------------------------
% 3.07/1.76 % (5004)------------------------------
% 3.07/1.77 % (5007)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/17Mi)
% 3.07/1.78 % (5007)Instruction limit reached!
% 3.07/1.78 % (5007)------------------------------
% 3.07/1.78 % (5007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.78 % (5007)Termination reason: Unknown
% 3.07/1.78 % (5007)Termination phase: shuffling
% 3.07/1.78
% 3.07/1.78 % (5007)Memory used [KB]: 1535
% 3.07/1.78 % (5007)Time elapsed: 0.010 s
% 3.07/1.78 % (5007)Instructions burned: 17 (million)
% 3.07/1.78 % (5007)------------------------------
% 3.07/1.78 % (5007)------------------------------
% 3.07/1.80 % (5009)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/3Mi)
% 3.07/1.80 % (5009)Instruction limit reached!
% 3.07/1.80 % (5009)------------------------------
% 3.07/1.80 % (5009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.80 % (5009)Termination reason: Unknown
% 3.07/1.80 % (5009)Termination phase: shuffling
% 3.07/1.80
% 3.07/1.80 % (5009)Memory used [KB]: 1279
% 3.07/1.80 % (5009)Time elapsed: 0.003 s
% 3.07/1.80 % (5009)Instructions burned: 3 (million)
% 3.07/1.80 % (5009)------------------------------
% 3.07/1.80 % (5009)------------------------------
% 3.07/1.82 % (5011)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/30Mi)
% 3.07/1.82 % (4993)Refutation not found, incomplete strategy
% 3.07/1.82 % (4993)------------------------------
% 3.07/1.82 % (4993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.82 % (4993)Termination reason: Refutation not found, incomplete strategy
% 3.07/1.82
% 3.07/1.82
% 3.07/1.82 % (4993)Memory used [KB]: 7291
% 3.07/1.82 % (4993)Time elapsed: 0.103 s
% 3.07/1.82 % (4993)Instructions burned: 214 (million)
% 3.07/1.82 % (4993)------------------------------
% 3.07/1.82 % (4993)------------------------------
% 3.07/1.83 % (5011)Instruction limit reached!
% 3.07/1.83 % (5011)------------------------------
% 3.07/1.83 % (5011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.83 % (5011)Termination reason: Unknown
% 3.07/1.83 % (5011)Termination phase: shuffling
% 3.07/1.83
% 3.07/1.83 % (5011)Memory used [KB]: 1535
% 3.07/1.83 % (5011)Time elapsed: 0.016 s
% 3.07/1.83 % (5011)Instructions burned: 30 (million)
% 3.07/1.83 % (5011)------------------------------
% 3.07/1.83 % (5011)------------------------------
% 3.07/1.84 % (5012)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/127Mi)
% 3.07/1.85 % (5013)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_4744 for (2998ds/100Mi)
% 3.07/1.87 % (5012)Instruction limit reached!
% 3.07/1.87 % (5012)------------------------------
% 3.07/1.87 % (5012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.87 % (5012)Termination reason: Unknown
% 3.07/1.87 % (5012)Termination phase: Saturation
% 3.07/1.87
% 3.07/1.87 % (5012)Memory used [KB]: 6524
% 3.07/1.87 % (5012)Time elapsed: 0.038 s
% 3.07/1.87 % (5012)Instructions burned: 129 (million)
% 3.07/1.87 % (5012)------------------------------
% 3.07/1.87 % (5012)------------------------------
% 3.07/1.88 % (5015)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_4744 for (2997ds/3Mi)
% 3.07/1.88 % (5015)Instruction limit reached!
% 3.07/1.88 % (5015)------------------------------
% 3.07/1.88 % (5015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.88 % (5015)Termination reason: Unknown
% 3.07/1.88 % (5015)Termination phase: shuffling
% 3.07/1.88
% 3.07/1.88 % (5015)Memory used [KB]: 1279
% 3.07/1.88 % (5015)Time elapsed: 0.002 s
% 3.07/1.88 % (5015)Instructions burned: 3 (million)
% 3.07/1.88 % (5015)------------------------------
% 3.07/1.88 % (5015)------------------------------
% 3.07/1.89 % (5013)Instruction limit reached!
% 3.07/1.89 % (5013)------------------------------
% 3.07/1.89 % (5013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.89 % (5013)Termination reason: Unknown
% 3.07/1.89 % (5013)Termination phase: Property scanning
% 3.07/1.89
% 3.07/1.89 % (5013)Memory used [KB]: 2174
% 3.07/1.89 % (5013)Time elapsed: 0.046 s
% 3.07/1.89 % (5013)Instructions burned: 102 (million)
% 3.07/1.89 % (5013)------------------------------
% 3.07/1.89 % (5013)------------------------------
% 3.07/1.90 % (5016)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_4744 for (2997ds/20Mi)
% 3.07/1.91 % (5016)Instruction limit reached!
% 3.07/1.91 % (5016)------------------------------
% 3.07/1.91 % (5016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.91 % (5016)Termination reason: Unknown
% 3.07/1.91 % (5016)Termination phase: shuffling
% 3.07/1.91
% 3.07/1.91 % (5016)Memory used [KB]: 1535
% 3.07/1.91 % (5016)Time elapsed: 0.012 s
% 3.07/1.91 % (5016)Instructions burned: 20 (million)
% 3.07/1.91 % (5016)------------------------------
% 3.07/1.91 % (5016)------------------------------
% 3.07/1.91 % (5017)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_4744 for (2997ds/86Mi)
% 3.07/1.92 % (5000)Instruction limit reached!
% 3.07/1.92 % (5000)------------------------------
% 3.07/1.92 % (5000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.92 % (5000)Termination reason: Unknown
% 3.07/1.92 % (5000)Termination phase: Saturation
% 3.07/1.92
% 3.07/1.92 % (5000)Memory used [KB]: 7419
% 3.07/1.92 % (5000)Time elapsed: 0.184 s
% 3.07/1.92 % (5000)Instructions burned: 379 (million)
% 3.07/1.92 % (5000)------------------------------
% 3.07/1.92 % (5000)------------------------------
% 3.07/1.92 % (5018)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_4744 for (2997ds/107Mi)
% 3.07/1.93 % (5021)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/39Mi)
% 3.07/1.94 % (5021)Instruction limit reached!
% 3.07/1.94 % (5021)------------------------------
% 3.07/1.94 % (5021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.94 % (5021)Termination reason: Unknown
% 3.07/1.94 % (5021)Termination phase: shuffling
% 3.07/1.94
% 3.07/1.94 % (5021)Memory used [KB]: 2046
% 3.07/1.94 % (5021)Time elapsed: 0.012 s
% 3.07/1.94 % (5021)Instructions burned: 41 (million)
% 3.07/1.94 % (5021)------------------------------
% 3.07/1.94 % (5021)------------------------------
% 3.07/1.95 % (5017)Instruction limit reached!
% 3.07/1.95 % (5017)------------------------------
% 3.07/1.95 % (5017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.95 % (5017)Termination reason: Unknown
% 3.07/1.95 % (5017)Termination phase: Property scanning
% 3.07/1.95
% 3.07/1.95 % (5017)Memory used [KB]: 2046
% 3.07/1.95 % (5017)Time elapsed: 0.041 s
% 3.07/1.95 % (5017)Instructions burned: 86 (million)
% 3.07/1.95 % (5017)------------------------------
% 3.07/1.95 % (5017)------------------------------
% 3.07/1.95 % (5023)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/448Mi)
% 3.07/1.97 % (5025)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/46Mi)
% 3.07/1.97 % (5018)Instruction limit reached!
% 3.07/1.97 % (5018)------------------------------
% 3.07/1.97 % (5018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.97 % (5018)Termination reason: Unknown
% 3.07/1.97 % (5018)Termination phase: Saturation
% 3.07/1.97
% 3.07/1.97 % (5018)Memory used [KB]: 6396
% 3.07/1.97 % (5018)Time elapsed: 0.050 s
% 3.07/1.97 % (5018)Instructions burned: 107 (million)
% 3.07/1.97 % (5018)------------------------------
% 3.07/1.97 % (5018)------------------------------
% 3.07/1.98 % (5001)Instruction limit reached!
% 3.07/1.98 % (5001)------------------------------
% 3.07/1.98 % (5001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.98 % (5001)Termination reason: Unknown
% 3.07/1.98 % (5001)Termination phase: Saturation
% 3.07/1.98
% 3.07/1.98 % (5001)Memory used [KB]: 8315
% 3.07/1.98 % (5001)Time elapsed: 0.240 s
% 3.07/1.98 % (5001)Instructions burned: 779 (million)
% 3.07/1.98 % (5001)------------------------------
% 3.07/1.98 % (5001)------------------------------
% 3.07/1.98 % (5025)Instruction limit reached!
% 3.07/1.98 % (5025)------------------------------
% 3.07/1.98 % (5025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.98 % (5025)Termination reason: Unknown
% 3.07/1.98 % (5025)Termination phase: SInE selection
% 3.07/1.98
% 3.07/1.98 % (5025)Memory used [KB]: 1535
% 3.07/1.98 % (5025)Time elapsed: 0.021 s
% 3.07/1.98 % (5025)Instructions burned: 49 (million)
% 3.07/1.98 % (5025)------------------------------
% 3.07/1.98 % (5025)------------------------------
% 3.07/1.99 % (5026)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/98Mi)
% 3.07/1.99 % (5027)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/507Mi)
% 3.07/1.99 % (5030)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_4744 for (2996ds/149Mi)
% 3.07/2.00 % (5005)First to succeed.
% 3.07/2.00 % (5005)Refutation found. Thanks to Tanya!
% 3.07/2.00 % SZS status Theorem for DTF2THF_4744
% 3.07/2.00 % SZS output start Proof for DTF2THF_4744
% 3.07/2.00 thf(type_def_5, type, partia1993116613t_unit: $tType).
% 3.07/2.00 thf(type_def_6, type, set_Product_prod_a_a: $tType).
% 3.07/2.00 thf(type_def_8, type, partia96731725t_unit: $tType).
% 3.07/2.00 thf(type_def_9, type, set_a: $tType).
% 3.07/2.00 thf(type_def_10, type, partia1833973666xt_a_b: $tType).
% 3.07/2.00 thf(type_def_11, type, a: $tType).
% 3.07/2.00 thf(type_def_12, type, int: $tType).
% 3.07/2.00 thf(type_def_13, type, nat: $tType).
% 3.07/2.00 thf(type_def_14, type, product_prod_a_a: $tType).
% 3.07/2.00 thf(type_def_15, type, b: $tType).
% 3.07/2.00 thf(func_def_0, type, partia1993116613t_unit: $tType).
% 3.07/2.00 thf(func_def_1, type, partia1833973666xt_a_b: $tType).
% 3.07/2.00 thf(func_def_2, type, partia96731725t_unit: $tType).
% 3.07/2.00 thf(func_def_3, type, set_Product_prod_a_a: $tType).
% 3.07/2.00 thf(func_def_4, type, product_prod_a_a: $tType).
% 3.07/2.00 thf(func_def_5, type, set_a: $tType).
% 3.07/2.00 thf(func_def_6, type, nat: $tType).
% 3.07/2.00 thf(func_def_7, type, int: $tType).
% 3.07/2.00 thf(func_def_8, type, b: $tType).
% 3.07/2.00 thf(func_def_9, type, a: $tType).
% 3.07/2.00 thf(func_def_10, type, partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a).
% 3.07/2.00 thf(func_def_11, type, partia1955795460t_unit: partia96731725t_unit > set_a).
% 3.07/2.00 thf(func_def_12, type, partia1066395285xt_a_b: partia1833973666xt_a_b > set_a).
% 3.07/2.00 thf(func_def_13, type, units_a_Product_unit: partia96731725t_unit > set_a).
% 3.07/2.00 thf(func_def_14, type, units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a).
% 3.07/2.00 thf(func_def_15, type, comm_g1684316527t_unit: partia96731725t_unit > $o).
% 3.07/2.00 thf(func_def_16, type, comm_g791708116xt_a_b: partia1833973666xt_a_b > $o).
% 3.07/2.00 thf(func_def_17, type, group_a_Product_unit: partia96731725t_unit > $o).
% 3.07/2.00 thf(func_def_18, type, group_a_ring_ext_a_b: partia1833973666xt_a_b > $o).
% 3.07/2.00 thf(func_def_19, type, mult_a_Product_unit: partia96731725t_unit > a > a > a).
% 3.07/2.00 thf(func_def_20, type, mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a).
% 3.07/2.00 thf(func_def_21, type, one_a_Product_unit: partia96731725t_unit > a).
% 3.07/2.00 thf(func_def_22, type, one_a_ring_ext_a_b: partia1833973666xt_a_b > a).
% 3.07/2.00 thf(func_def_23, type, units_873712258t_unit: partia96731725t_unit > partia96731725t_unit).
% 3.07/2.00 thf(func_def_24, type, units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit).
% 3.07/2.00 thf(func_def_25, type, minus_minus_int: int > int > int).
% 3.07/2.00 thf(func_def_26, type, minus_minus_nat: nat > nat > nat).
% 3.07/2.00 thf(func_def_27, type, plus_plus_int: int > int > int).
% 3.07/2.00 thf(func_def_28, type, plus_plus_nat: nat > nat > nat).
% 3.07/2.00 thf(func_def_29, type, uminus_uminus_int: int > int).
% 3.07/2.00 thf(func_def_30, type, suc: nat > nat).
% 3.07/2.00 thf(func_def_31, type, semiri2019852685at_int: nat > int).
% 3.07/2.00 thf(func_def_32, type, semiri1382578993at_nat: nat > nat).
% 3.07/2.00 thf(func_def_33, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.07/2.00 thf(func_def_34, type, a_inv_a_b: partia1833973666xt_a_b > a > a).
% 3.07/2.00 thf(func_def_35, type, a_minus_a_b: partia1833973666xt_a_b > a > a > a).
% 3.07/2.00 thf(func_def_36, type, abelian_group_a_b: partia1833973666xt_a_b > $o).
% 3.07/2.00 thf(func_def_37, type, add_pow_a_b_int: partia1833973666xt_a_b > int > a > a).
% 3.07/2.00 thf(func_def_38, type, add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a).
% 3.07/2.00 thf(func_def_39, type, field_a_b: partia1833973666xt_a_b > $o).
% 3.07/2.00 thf(func_def_40, type, add_a_b: partia1833973666xt_a_b > a > a > a).
% 3.07/2.00 thf(func_def_41, type, more_a_b: partia1833973666xt_a_b > b).
% 3.07/2.00 thf(func_def_42, type, zero_a_b: partia1833973666xt_a_b > a).
% 3.07/2.00 thf(func_def_43, type, semiring_a_b: partia1833973666xt_a_b > $o).
% 3.07/2.00 thf(func_def_44, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 3.07/2.00 thf(func_def_45, type, collect_a: (a > $o) > set_a).
% 3.07/2.00 thf(func_def_46, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.07/2.00 thf(func_def_47, type, member_a: a > set_a > $o).
% 3.07/2.00 thf(func_def_48, type, r: partia1833973666xt_a_b).
% 3.07/2.00 thf(func_def_49, type, r2: a).
% 3.07/2.00 thf(func_def_50, type, r3: a).
% 3.07/2.00 thf(func_def_51, type, r4: a).
% 3.07/2.00 thf(func_def_52, type, rel: partia1993116613t_unit).
% 3.07/2.00 thf(func_def_53, type, s: a).
% 3.07/2.00 thf(func_def_54, type, s2: a).
% 3.07/2.00 thf(func_def_55, type, s3: a).
% 3.07/2.00 thf(func_def_65, type, sK0: (product_prod_a_a > $o) > a).
% 3.07/2.00 thf(func_def_66, type, sK1: (product_prod_a_a > $o) > a).
% 3.07/2.00 thf(func_def_67, type, sK2: product_prod_a_a > a).
% 3.07/2.00 thf(func_def_68, type, sK3: product_prod_a_a > a).
% 3.07/2.00 thf(func_def_69, type, sK4: product_prod_a_a > a).
% 3.07/2.00 thf(func_def_70, type, sK5: product_prod_a_a > a).
% 3.07/2.00 thf(func_def_71, type, sK6: (product_prod_a_a > $o) > (product_prod_a_a > $o) > product_prod_a_a).
% 3.07/2.00 thf(func_def_72, type, sK7: (product_prod_a_a > $o) > a).
% 3.07/2.00 thf(func_def_73, type, sK8: (product_prod_a_a > $o) > a).
% 3.07/2.00 thf(func_def_76, type, ph10: !>[X0: $tType]:(X0)).
% 3.07/2.00 thf(f1800,plain,(
% 3.07/2.00 $false),
% 3.07/2.00 inference(subsumption_resolution,[status(thm)],[f1799,f787])).
% 3.07/2.00 thf(f787,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s3))) @ (partia206007992t_unit @ rel)) != $true)),
% 3.07/2.00 inference(backward_demodulation,[status(thm)],[f739,f731])).
% 3.07/2.00 thf(f731,plain,(
% 3.07/2.00 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s3)))),
% 3.07/2.00 inference(cnf_transformation,[status(thm)],[f7])).
% 3.07/2.00 thf(f7,axiom,(
% 3.07/2.00 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s3)))),
% 3.07/2.00 file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_6_f9)).
% 3.07/2.00 thf(f739,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel)) != $true)),
% 3.07/2.00 inference(cnf_transformation,[status(thm)],[f664])).
% 3.07/2.00 thf(f664,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel)) != $true)),
% 3.07/2.00 inference(flattening,[status(thm)],[f522])).
% 3.07/2.00 thf(f522,plain,(
% 3.07/2.00 ~((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(fool_elimination,[status(thm)],[f521])).
% 3.07/2.00 thf(f521,plain,(
% 3.07/2.00 ~(member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 inference(rectify,[status(thm)],[f284])).
% 3.07/2.00 thf(f284,negated_conjecture,(
% 3.07/2.00 ~(member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 inference(negated_conjecture,[status(cth)],[f283])).
% 3.07/2.00 thf(f283,conjecture,(
% 3.07/2.00 (member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',conj_0)).
% 3.07/2.00 thf(f1799,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s3))) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(forward_demodulation,[status(thm)],[f1796,f731])).
% 3.07/2.00 thf(f1796,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(trivial_inequality_removal,[status(thm)],[f1791])).
% 3.07/2.00 thf(f1791,plain,(
% 3.07/2.00 ($true != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ (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 @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ r3 @ r4))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3))) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(superposition,[status(thm)],[f1222,f736])).
% 3.07/2.00 thf(f736,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r3 @ r4) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(cnf_transformation,[status(thm)],[f336])).
% 3.07/2.00 thf(f336,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r3 @ r4) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(fool_elimination,[status(thm)],[f335])).
% 3.07/2.00 thf(f335,plain,(
% 3.07/2.00 (member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r3 @ r4) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 inference(rectify,[status(thm)],[f11])).
% 3.07/2.00 thf(f11,axiom,(
% 3.07/2.00 (member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r3 @ r4) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_10_f5)).
% 3.07/2.00 thf(f1222,plain,(
% 3.07/2.00 ( ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ (partia206007992t_unit @ rel)) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X1 @ (mult_a_ring_ext_a_b @ r @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ X0)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ X1)) @ (partia206007992t_unit @ rel)) = $true)) )),
% 3.07/2.00 inference(trivial_inequality_removal,[status(thm)],[f1217])).
% 3.07/2.00 thf(f1217,plain,(
% 3.07/2.00 ( ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ (partia206007992t_unit @ rel)) != $true) | ($true != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X1 @ (mult_a_ring_ext_a_b @ r @ r2 @ r4)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ X0)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s @ s3) @ X1)) @ (partia206007992t_unit @ rel)) = $true)) )),
% 3.07/2.00 inference(superposition,[status(thm)],[f715,f716])).
% 3.07/2.00 thf(f716,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r2 @ r4) @ (mult_a_ring_ext_a_b @ r @ s @ s3)) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(cnf_transformation,[status(thm)],[f476])).
% 3.07/2.00 thf(f476,plain,(
% 3.07/2.00 ((member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r2 @ r4) @ (mult_a_ring_ext_a_b @ r @ s @ s3)) @ (partia206007992t_unit @ rel)) = $true)),
% 3.07/2.00 inference(fool_elimination,[status(thm)],[f475])).
% 3.07/2.00 thf(f475,plain,(
% 3.07/2.00 (member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r2 @ r4) @ (mult_a_ring_ext_a_b @ r @ s @ s3)) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 inference(rectify,[status(thm)],[f12])).
% 3.07/2.00 thf(f12,axiom,(
% 3.07/2.00 (member449909584od_a_a @ (product_Pair_a_a @ (mult_a_ring_ext_a_b @ r @ r2 @ r4) @ (mult_a_ring_ext_a_b @ r @ s @ s3)) @ (partia206007992t_unit @ rel))),
% 3.07/2.00 file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_11_f4)).
% 3.07/2.00 thf(f715,plain,(
% 3.07/2.00 ( ! [X2 : a,X3 : a,X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ (partia206007992t_unit @ rel)) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X2) @ (partia206007992t_unit @ rel)) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X2 @ X3) @ (mult_a_ring_ext_a_b @ r @ X0 @ X1)) @ (mult_a_ring_ext_a_b @ r @ X0 @ X2)) @ (partia206007992t_unit @ rel)) = $true)) )),
% 3.07/2.00 inference(cnf_transformation,[status(thm)],[f684])).
% 3.07/2.00 thf(f684,plain,(
% 3.07/2.00 ! [X0 : a,X1 : a,X2 : a,X3 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ (partia206007992t_unit @ rel)) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X2 @ X3) @ (mult_a_ring_ext_a_b @ r @ X0 @ X1)) @ (mult_a_ring_ext_a_b @ r @ X0 @ X2)) @ (partia206007992t_unit @ rel)) = $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X2) @ (partia206007992t_unit @ rel)) != $true))),
% 3.07/2.00 inference(flattening,[status(thm)],[f683])).
% 3.07/2.00 thf(f683,plain,(
% 3.07/2.00 ! [X0 : a,X1 : a,X2 : a,X3 : a] : ((((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X2 @ X3) @ (mult_a_ring_ext_a_b @ r @ X0 @ X1)) @ (mult_a_ring_ext_a_b @ r @ X0 @ X2)) @ (partia206007992t_unit @ rel)) = $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X2) @ (partia206007992t_unit @ rel)) != $true)) | ((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ (partia206007992t_unit @ rel)) != $true))),
% 3.07/2.00 inference(ennf_transformation,[status(thm)],[f474])).
% 3.07/2.00 thf(f474,plain,(
% 3.07/2.00 ! [X0 : a,X1 : a,X2 : a,X3 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ (partia206007992t_unit @ rel)) = $true) => (((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X2) @ (partia206007992t_unit @ rel)) = $true) => ((member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X2 @ X3) @ (mult_a_ring_ext_a_b @ r @ X0 @ X1)) @ (mult_a_ring_ext_a_b @ r @ X0 @ X2)) @ (partia206007992t_unit @ rel)) = $true)))),
% 3.07/2.00 inference(fool_elimination,[status(thm)],[f473])).
% 3.07/2.00 thf(f473,plain,(
% 3.07/2.00 ! [X0 : a,X1 : a,X2 : a,X3 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ (partia206007992t_unit @ rel)) => ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X2) @ (partia206007992t_unit @ rel)) => (member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X2 @ X3) @ (mult_a_ring_ext_a_b @ r @ X0 @ X1)) @ (mult_a_ring_ext_a_b @ r @ X0 @ X2)) @ (partia206007992t_unit @ rel))))),
% 3.07/2.00 inference(rectify,[status(thm)],[f5])).
% 3.07/2.00 thf(f5,axiom,(
% 3.07/2.00 ! [X4 : a,X5 : a,X6 : a,X3 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X4) @ (partia206007992t_unit @ rel)) => ((member449909584od_a_a @ (product_Pair_a_a @ X5 @ X6) @ (partia206007992t_unit @ rel)) => (member449909584od_a_a @ (product_Pair_a_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ X6 @ X3) @ (mult_a_ring_ext_a_b @ r @ X4 @ X5)) @ (mult_a_ring_ext_a_b @ r @ X4 @ X6)) @ (partia206007992t_unit @ rel))))),
% 3.07/2.00 file('/export/starexec/sandbox/tmp/tmp.2gzgM4eQP7/DTF2THF_4744.p',fact_4_closed__rel__add)).
% 3.07/2.00 % SZS output end Proof for DTF2THF_4744
% 3.07/2.00 % (5005)------------------------------
% 3.07/2.00 % (5005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/2.00 % (5005)Termination reason: Refutation
% 3.07/2.00
% 3.07/2.00 % (5005)Memory used [KB]: 6908
% 3.07/2.00 % (5005)Time elapsed: 0.249 s
% 3.07/2.00 % (5005)Instructions burned: 505 (million)
% 3.07/2.00 % (5005)------------------------------
% 3.07/2.00 % (5005)------------------------------
% 3.07/2.00 % (4951)Success in time 0.382 s
%------------------------------------------------------------------------------