%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP108^1 : 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 : n015.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 2.70s 1.65s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 13
% Syntax : Number of formulae : 89 ( 87 unt; 0 typ; 0 def)
% Number of atoms : 215 ( 98 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 5346 ( 8 ~; 0 |; 0 &;5296 @)
% ( 0 <=>; 38 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 38 usr; 14 con; 0-3 aty)
% ( 4 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 97 ( 93 ^; 4 !; 0 ?; 97 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
partia1833973666xt_a_b: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_7,type,
set_a: $tType ).
thf(type_def_9,type,
int: $tType ).
thf(type_def_10,type,
product_prod_a_a: $tType ).
thf(func_def_0,type,
partia1833973666xt_a_b: $tType ).
thf(func_def_1,type,
product_prod_a_a: $tType ).
thf(func_def_2,type,
set_a: $tType ).
thf(func_def_3,type,
int: $tType ).
thf(func_def_4,type,
a: $tType ).
thf(func_def_5,type,
a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
thf(func_def_6,type,
additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
thf(func_def_7,type,
set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
thf(func_def_8,type,
partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
thf(func_def_9,type,
mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_10,type,
minus_minus_int: int > int > int ).
thf(func_def_11,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_12,type,
plus_plus_int: int > int > int ).
thf(func_def_13,type,
uminus_uminus_int: int > int ).
thf(func_def_14,type,
uminus_uminus_set_a: set_a > set_a ).
thf(func_def_15,type,
locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_16,type,
locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_17,type,
locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_18,type,
locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_19,type,
bot_bot_set_a: set_a ).
thf(func_def_20,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_21,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_22,type,
product_fst_a_a: product_prod_a_a > a ).
thf(func_def_23,type,
product_snd_a_a: product_prod_a_a > a ).
thf(func_def_24,type,
a_inv_a_b: partia1833973666xt_a_b > a > a ).
thf(func_def_25,type,
a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_26,type,
add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
thf(func_def_27,type,
add_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_28,type,
zero_a_b: partia1833973666xt_a_b > a ).
thf(func_def_29,type,
semiring_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_30,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_31,type,
member_a: a > set_a > $o ).
thf(func_def_32,type,
r: partia1833973666xt_a_b ).
thf(func_def_33,type,
s: set_a ).
thf(func_def_34,type,
r2: a ).
thf(func_def_35,type,
r3: a ).
thf(func_def_36,type,
s2: a ).
thf(func_def_37,type,
s3: a ).
thf(func_def_38,type,
t: a ).
thf(func_def_39,type,
t2: a ).
thf(func_def_40,type,
x: product_prod_a_a ).
thf(func_def_41,type,
y: product_prod_a_a ).
thf(func_def_58,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f1246,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1245]) ).
thf(f1245,plain,
$false = $true,
inference(boolean_simplification,[status(thm)],[f1244]) ).
thf(f1244,plain,
( ( ~ $true ) = $true ),
inference(backward_demodulation,[status(thm)],[f1115,f1243]) ).
thf(f1243,plain,
( ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1242]) ).
thf(f1242,plain,
( ( $true
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1134,f1233]) ).
thf(f1233,plain,
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1224]) ).
thf(f1224,plain,
( ( $true
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) ) )
= $true ),
inference(superposition,[status(thm)],[f1206,f781]) ).
thf(f781,plain,
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s )
= $true ),
inference(cnf_transformation,[status(thm)],[f356]) ).
thf(f356,plain,
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s )
= $true ),
inference(fool_elimination,[status(thm)],[f355]) ).
thf(f355,plain,
member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s,
inference(rectify,[status(thm)],[f66]) ).
thf(f66,axiom,
member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s,
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_65_f8) ).
thf(f1206,plain,
! [X1: a] :
( ( ( member_a @ X1 @ s )
=> ( member_a @ X1 @ ( partia1066395285xt_a_b @ r ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1205]) ).
thf(f1205,plain,
! [X1: a] :
( $true
= ( ^ [Y0: a] :
( ( member_a @ Y0 @ s )
=> ( member_a @ Y0 @ ( partia1066395285xt_a_b @ r ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1072]) ).
thf(f1072,plain,
( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ s )
=> ( member_a @ Y0 @ ( partia1066395285xt_a_b @ r ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1008]) ).
thf(f1008,plain,
( $true
= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ s
@ ( partia1066395285xt_a_b @ r ) ) ),
inference(definition_unfolding,[status(thm)],[f884,f857]) ).
thf(f857,plain,
( ord_less_eq_set_a
= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f631]) ).
thf(f631,plain,
( ord_less_eq_set_a
= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f630]) ).
thf(f630,plain,
( ( ^ [X0: set_a,X1: set_a] :
! [X2: a] :
( ( member_a @ X2 @ X0 )
=> ( member_a @ X2 @ X1 ) ) )
= ord_less_eq_set_a ),
inference(rectify,[status(thm)],[f244]) ).
thf(f244,axiom,
( ( ^ [X44: set_a,X45: set_a] :
! [X9: a] :
( ( member_a @ X9 @ X44 )
=> ( member_a @ X9 @ X45 ) ) )
= ord_less_eq_set_a ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_243_subset__eq) ).
thf(f884,plain,
( ( ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f648]) ).
thf(f648,plain,
( ( ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(fool_elimination,[status(thm)],[f647]) ).
thf(f647,plain,
ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ),
inference(rectify,[status(thm)],[f143]) ).
thf(f143,axiom,
ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_142_subset) ).
thf(f1134,plain,
( ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1128,f1133]) ).
thf(f1133,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1132,f1074]) ).
thf(f1074,plain,
( ( zero_a_b @ r )
= ( mult_a_ring_ext_a_b @ r @ t @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1022]) ).
thf(f1022,plain,
( ( zero_a_b @ r )
= ( mult_a_ring_ext_a_b @ r @ t
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 )
@ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f934,f890]) ).
thf(f890,plain,
( a_minus_a_b
= ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f577]) ).
thf(f577,plain,
( a_minus_a_b
= ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f576]) ).
thf(f576,plain,
( ( ^ [X0: partia1833973666xt_a_b,X1: a,X2: a] : ( add_a_b @ X0 @ X1 @ ( a_inv_a_b @ X0 @ X2 ) ) )
= a_minus_a_b ),
inference(rectify,[status(thm)],[f131]) ).
thf(f131,axiom,
( ( ^ [X30: partia1833973666xt_a_b,X9: a,X31: a] : ( add_a_b @ X30 @ X9 @ ( a_inv_a_b @ X30 @ X31 ) ) )
= a_minus_a_b ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_130_a__minus__def) ).
thf(f934,plain,
( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
= ( zero_a_b @ r ) ),
inference(cnf_transformation,[status(thm)],[f361]) ).
thf(f361,plain,
( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
= ( zero_a_b @ r ) ),
inference(fool_elimination,[status(thm)],[f59]) ).
thf(f59,axiom,
( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
= ( zero_a_b @ r ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_58_f7) ).
thf(f1132,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1131,f1073]) ).
thf(f1073,plain,
( ( zero_a_b @ r )
= ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1014]) ).
thf(f1014,plain,
( ( zero_a_b @ r )
= ( mult_a_ring_ext_a_b @ r @ t2
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 )
@ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f910,f890]) ).
thf(f910,plain,
( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
= ( zero_a_b @ r ) ),
inference(cnf_transformation,[status(thm)],[f305]) ).
thf(f305,plain,
( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
= ( zero_a_b @ r ) ),
inference(fool_elimination,[status(thm)],[f58]) ).
thf(f58,axiom,
( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
= ( zero_a_b @ r ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_57_f5) ).
thf(f1131,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1042,f1120]) ).
thf(f1120,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1071,f1119]) ).
thf(f1119,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f1107,f1073]) ).
thf(f1107,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f990]) ).
thf(f990,plain,
( ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) )
@ ( mult_a_ring_ext_a_b @ r @ t2
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 )
@ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f848,f890,f890]) ).
thf(f848,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f312]) ).
thf(f312,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f42]) ).
thf(f42,axiom,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_41_f25) ).
thf(f1071,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f972]) ).
thf(f972,plain,
( ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f776,f890,f890]) ).
thf(f776,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f290]) ).
thf(f290,plain,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f43]) ).
thf(f43,axiom,
( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
= ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_42_f32) ).
thf(f1042,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f971]) ).
thf(f971,plain,
( ( add_a_b @ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) )
@ ( mult_a_ring_ext_a_b @ r @ t2
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 )
@ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) )
@ ( mult_a_ring_ext_a_b @ r @ t
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 )
@ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( add_a_b @ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f768,f890,f890,f890,f890]) ).
thf(f768,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f344]) ).
thf(f344,plain,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f20]) ).
thf(f20,axiom,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_19_f40) ).
thf(f1128,plain,
( ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1127,f1120]) ).
thf(f1127,plain,
( ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1126]) ).
thf(f1126,plain,
( ( $true
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1125,f1081]) ).
thf(f1081,plain,
( $true
= ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ) ),
inference(beta_eta_normalization,[status(thm)],[f980]) ).
thf(f980,plain,
( ( member_a
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
@ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f805,f890]) ).
thf(f805,plain,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f255]) ).
thf(f255,plain,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(fool_elimination,[status(thm)],[f254]) ).
thf(f254,plain,
member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_13_f42) ).
thf(f1125,plain,
( ( ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1124]) ).
thf(f1124,plain,
( ( $true
=> ( ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1056,f1062]) ).
thf(f1062,plain,
( ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f946]) ).
thf(f946,plain,
( $true
= ( member_a
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
@ ( partia1066395285xt_a_b @ r ) ) ),
inference(definition_unfolding,[status(thm)],[f701,f890]) ).
thf(f701,plain,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f261]) ).
thf(f261,plain,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
= $true ),
inference(fool_elimination,[status(thm)],[f260]) ).
thf(f260,plain,
member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ),
inference(rectify,[status(thm)],[f13]) ).
thf(f13,axiom,
member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_12_f41) ).
thf(f1056,plain,
( ( ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f995]) ).
thf(f995,plain,
( ( ( member_a
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
@ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
@ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( add_a_b @ r
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f860,f890,f890,f890,f890,f890,f890]) ).
thf(f860,plain,
( ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f570]) ).
thf(f570,plain,
( ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f569]) ).
thf(f569,plain,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f22]) ).
thf(f22,axiom,
( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062) ).
thf(f1115,plain,
( ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( zero_a_b @ r ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1114,f1073]) ).
thf(f1114,plain,
( ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( zero_a_b @ r ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1076,f1074]) ).
thf(f1076,plain,
( $true
= ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1020]) ).
thf(f1020,plain,
( ( ( add_a_b @ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) )
@ ( mult_a_ring_ext_a_b @ r @ t2
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 )
@ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) )
@ ( mult_a_ring_ext_a_b @ r @ t
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 )
@ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t )
@ ( add_a_b @ r
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
@ ( ^ [Y0: partia1833973666xt_a_b,Y1: a,Y2: a] : ( add_a_b @ Y0 @ Y1 @ ( a_inv_a_b @ Y0 @ Y2 ) )
@ r
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) )
@ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f928,f890,f890,f890,f890]) ).
thf(f928,plain,
( ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f420]) ).
thf(f420,plain,
( ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f246]) ).
thf(f246,negated_conjecture,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
!= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f245]) ).
thf(f245,conjecture,
( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP108^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n015.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 08:26:16 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
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP108^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer Localization problem prob_758__8980450_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : Localization/prob_758__8980450_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 287 ( 119 unt; 42 typ; 0 def)
% 0.19/0.45 % Number of atoms : 568 ( 250 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 8 ( 2 avg)
% 0.19/0.45 % Number of connectives : 3894 ( 18 ~; 0 |; 8 &;3625 @)
% 0.19/0.45 % ( 0 <=>; 243 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 15 ( 6 avg)
% 0.19/0.45 % Number of types : 6 ( 5 usr)
% 0.19/0.45 % Number of type conns : 58 ( 58 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 38 ( 37 usr; 11 con; 0-3 aty)
% 0.19/0.45 % Number of variables : 464 ( 16 ^; 445 !; 3 ?; 464 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:39:18.132
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (5)
% 0.19/0.45 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,
% 0.19/0.45 partia1833973666xt_a_b: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.19/0.45 product_prod_a_a: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.19/0.45 set_a: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Int__Oint,type,
% 0.19/0.45 int: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_tf__a,type,
% 0.19/0.45 a: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (37)
% 0.19/0.45 thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
% 0.19/0.45 a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
% 0.19/0.45 additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
% 0.19/0.45 set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
% 0.19/0.45
% 0.19/0.45 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,
% 0.19/0.45 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.19/0.45 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.19/0.45 minus_minus_int: int > int > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.45 minus_minus_set_a: set_a > set_a > set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.19/0.45 plus_plus_int: int > int > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.19/0.45 uminus_uminus_int: int > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.45 uminus_uminus_set_a: set_a > set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oeq__obj__rng__of__frac_001tf__a_001tf__b,type,
% 0.19/0.45 locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__crng_001tf__a_001tf__b,type,
% 0.19/0.45 locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__rng_001tf__a_001tf__b,type,
% 0.19/0.45 locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Osubmonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.19/0.45 locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.45 bot_bot_set_a: set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.19/0.45 ord_less_eq_int: int > int > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.45 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
% 0.19/0.45 product_fst_a_a: product_prod_a_a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
% 0.19/0.45 product_snd_a_a: product_prod_a_a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 0.19/0.45 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 0.19/0.45 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 0.19/0.45 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 0.19/0.45 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 0.19/0.45 zero_a_b: partia1833973666xt_a_b > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 0.19/0.45 semiring_a_b: partia1833973666xt_a_b > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.19/0.45 collect_a: ( a > $o ) > set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001tf__a,type,
% 0.19/0.45 member_a: a > set_a > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_R,type,
% 0.19/0.45 r: partia1833973666xt_a_b ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_S,type,
% 0.19/0.45 s: set_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_r,type,
% 0.19/0.45 r2: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_r_H,type,
% 0.19/0.45 r3: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_s,type,
% 0.19/0.45 s2: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_s_H,type,
% 0.19/0.45 s3: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_t_H____,type,
% 0.19/0.45 t: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_t____,type,
% 0.19/0.45 t2: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_x_H____,type,
% 0.19/0.45 x: product_prod_a_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_y_H____,type,
% 0.19/0.45 y: product_prod_a_a ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (244)
% 0.19/0.45 thf(fact_0_f18,axiom,
% 0.19/0.45 member_a @ t @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f18
% 0.19/0.45 thf(fact_1_f17,axiom,
% 0.19/0.45 member_a @ t2 @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f17
% 0.19/0.45 thf(fact_2_right__add__eq,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( add_a_b @ r @ C @ A )
% 0.19/0.45 = ( add_a_b @ r @ C @ B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % right_add_eq
% 0.19/0.45 thf(fact_3_local_Oright__minus__eq,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( a_minus_a_b @ r @ C @ A )
% 0.19/0.45 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.right_minus_eq
% 0.19/0.45 thf(fact_4_m__assoc,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % m_assoc
% 0.19/0.45 thf(fact_5_m__comm,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % m_comm
% 0.19/0.45 thf(fact_6_m__lcomm,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % m_lcomm
% 0.19/0.45 thf(fact_7_add_Om__assoc,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.19/0.45 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.m_assoc
% 0.19/0.45 thf(fact_8_add_Om__comm,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ X @ Y )
% 0.19/0.45 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.m_comm
% 0.19/0.45 thf(fact_9_add_Om__lcomm,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 0.19/0.45 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.m_lcomm
% 0.19/0.45 thf(fact_10_f20,axiom,
% 0.19/0.45 member_a @ s3 @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f20
% 0.19/0.45 thf(fact_11_f19,axiom,
% 0.19/0.45 member_a @ s2 @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f19
% 0.19/0.45 thf(fact_12_f41,axiom,
% 0.19/0.45 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f41
% 0.19/0.45 thf(fact_13_f42,axiom,
% 0.19/0.45 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f42
% 0.19/0.45 thf(fact_14_l__distr,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_distr
% 0.19/0.45 thf(fact_15_r__distr,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_distr
% 0.19/0.45 thf(fact_16_four__elem__comm,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a,D: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % four_elem_comm
% 0.19/0.45 thf(fact_17_right__inv__add,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 0.19/0.45 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % right_inv_add
% 0.19/0.45 thf(fact_18_f21,axiom,
% 0.19/0.45 member_a @ ( product_snd_a_a @ y ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f21
% 0.19/0.45 thf(fact_19_f40,axiom,
% 0.19/0.45 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f40
% 0.19/0.45 thf(fact_20_f22,axiom,
% 0.19/0.45 member_a @ ( product_fst_a_a @ x ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f22
% 0.19/0.45 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.19/0.45 ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>\<lbrakk>snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<in> carrier R; snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y') \<in> carrier R; t \<otimes> t' \<in> carrier R\<rbrakk> \<Longrightarrow> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<oplus> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.19/0.45 thf(fact_22_f15,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f15
% 0.19/0.45 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>snd x' \<otimes> snd y' \<in> carrier R\<close>
% 0.19/0.45 thf(fact_24_f28,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f28
% 0.19/0.45 thf(fact_25_f11,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f11
% 0.19/0.45 thf(fact_26_f10,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f10
% 0.19/0.45 thf(fact_27_f27,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f27
% 0.19/0.45 thf(fact_28_f35,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f35
% 0.19/0.45 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = t \<otimes> (snd x' \<otimes> r) \<ominus> t \<otimes> (s \<otimes> fst x')\<close>
% 0.19/0.45 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = t' \<otimes> (snd y' \<otimes> r') \<ominus> t' \<otimes> (s' \<otimes> fst y')\<close>
% 0.19/0.45 thf(fact_31_f16,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f16
% 0.19/0.45 thf(fact_32_f24,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f24
% 0.19/0.45 thf(fact_33_f9,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f9
% 0.19/0.45 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t \<otimes> s \<otimes> snd x' \<in> carrier R\<close>
% 0.19/0.45 thf(fact_35_f23,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f23
% 0.19/0.45 thf(fact_36_f34,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f34
% 0.19/0.45 thf(fact_37_f36,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f36
% 0.19/0.45 thf(fact_38_f13,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f13
% 0.19/0.45 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.19/0.45 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) = t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (snd x' \<otimes> r) \<ominus> t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (s \<otimes> fst x')\<close>
% 0.19/0.45 thf(fact_40_f12,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f12
% 0.19/0.45 thf(fact_41_f25,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f25
% 0.19/0.45 thf(fact_42_f32,axiom,
% 0.19/0.45 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f32
% 0.19/0.45 thf(fact_43_f26,axiom,
% 0.19/0.45 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f26
% 0.19/0.45 thf(fact_44_f29,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f29
% 0.19/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.45 ! [A: a,P: a > $o] :
% 0.19/0.45 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.19/0.45 = ( P @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 % mem_Collect_eq
% 0.19/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.19/0.45 ! [A2: set_a] :
% 0.19/0.45 ( ( collect_a
% 0.19/0.45 @ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
% 0.19/0.45 = A2 ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_mem_eq
% 0.19/0.45 thf(fact_47_Collect__cong,axiom,
% 0.19/0.45 ! [P: a > $o,Q: a > $o] :
% 0.19/0.45 ( ! [X3: a] :
% 0.19/0.45 ( ( P @ X3 )
% 0.19/0.45 = ( Q @ X3 ) )
% 0.19/0.45 => ( ( collect_a @ P )
% 0.19/0.45 = ( collect_a @ Q ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_cong
% 0.19/0.45 thf(fact_48_f30,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f30
% 0.19/0.45 thf(fact_49_f33,axiom,
% 0.19/0.45 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 0.19/0.45 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f33
% 0.19/0.45 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s' \<otimes> r) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s \<otimes> r')\<close>
% 0.19/0.45 thf(fact_51_f14,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f14
% 0.19/0.45 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.19/0.45 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r')) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.19/0.45 thf(fact_53_f31,axiom,
% 0.19/0.45 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.19/0.45 = ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % f31
% 0.19/0.45 thf(fact_54_f37,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f37
% 0.19/0.45 thf(fact_55_f39,axiom,
% 0.19/0.45 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f39
% 0.19/0.45 thf(fact_56_f38,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % f38
% 0.19/0.45 thf(fact_57_f5,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.19/0.45 = ( zero_a_b @ r ) ) ).
% 0.19/0.45
% 0.19/0.45 % f5
% 0.19/0.45 thf(fact_58_f7,axiom,
% 0.19/0.45 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.19/0.45 = ( zero_a_b @ r ) ) ).
% 0.19/0.45
% 0.19/0.45 % f7
% 0.19/0.45 thf(fact_59_minus__closed,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_closed
% 0.19/0.45 thf(fact_60_add_Om__closed,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.m_closed
% 0.19/0.45 thf(fact_61_local_Oadd_Oright__cancel,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ Y @ X )
% 0.19/0.45 = ( add_a_b @ r @ Z @ X ) )
% 0.19/0.45 = ( Y = Z ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.add.right_cancel
% 0.19/0.45 thf(fact_62_semiring__simprules_I3_J,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % semiring_simprules(3)
% 0.19/0.45 thf(fact_63_f4,axiom,
% 0.19/0.45 member_a @ t2 @ s ).
% 0.19/0.45
% 0.19/0.45 % f4
% 0.19/0.45 thf(fact_64_f6,axiom,
% 0.19/0.45 member_a @ t @ s ).
% 0.19/0.45
% 0.19/0.45 % f6
% 0.19/0.45 thf(fact_65_f8,axiom,
% 0.19/0.45 member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s ).
% 0.19/0.45
% 0.19/0.45 % f8
% 0.19/0.45 thf(fact_66_local_Osemiring__axioms,axiom,
% 0.19/0.45 semiring_a_b @ r ).
% 0.19/0.45
% 0.19/0.45 % local.semiring_axioms
% 0.19/0.45 thf(fact_67_minus__eq,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( a_minus_a_b @ r @ X @ Y )
% 0.19/0.45 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_eq
% 0.19/0.45 thf(fact_68_prod__eqI,axiom,
% 0.19/0.45 ! [P2: product_prod_a_a,Q2: product_prod_a_a] :
% 0.19/0.45 ( ( ( product_fst_a_a @ P2 )
% 0.19/0.45 = ( product_fst_a_a @ Q2 ) )
% 0.19/0.45 => ( ( ( product_snd_a_a @ P2 )
% 0.19/0.45 = ( product_snd_a_a @ Q2 ) )
% 0.19/0.45 => ( P2 = Q2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % prod_eqI
% 0.19/0.45 thf(fact_69_exE__realizer_H,axiom,
% 0.19/0.45 ! [P: a > a > $o,P2: product_prod_a_a] :
% 0.19/0.45 ( ( P @ ( product_snd_a_a @ P2 ) @ ( product_fst_a_a @ P2 ) )
% 0.19/0.45 => ~ ! [X3: a,Y2: a] :
% 0.19/0.45 ~ ( P @ Y2 @ X3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % exE_realizer'
% 0.19/0.45 thf(fact_70_prod_Oexpand,axiom,
% 0.19/0.45 ! [Prod: product_prod_a_a,Prod2: product_prod_a_a] :
% 0.19/0.45 ( ( ( ( product_fst_a_a @ Prod )
% 0.19/0.45 = ( product_fst_a_a @ Prod2 ) )
% 0.19/0.45 & ( ( product_snd_a_a @ Prod )
% 0.19/0.45 = ( product_snd_a_a @ Prod2 ) ) )
% 0.19/0.45 => ( Prod = Prod2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % prod.expand
% 0.19/0.45 thf(fact_71_local_Ominus__unique,axiom,
% 0.19/0.45 ! [Y: a,X: a,Y3: a] :
% 0.19/0.45 ( ( ( add_a_b @ r @ Y @ X )
% 0.19/0.45 = ( zero_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ X @ Y3 )
% 0.19/0.45 = ( zero_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.minus_unique
% 0.19/0.45 thf(fact_72_add_Or__inv__ex,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ? [X3: a] :
% 0.19/0.45 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 & ( ( add_a_b @ r @ X @ X3 )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.r_inv_ex
% 0.19/0.45 thf(fact_73_add_Oone__unique,axiom,
% 0.19/0.45 ! [U: a] :
% 0.19/0.45 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ! [X3: a] :
% 0.19/0.45 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ U @ X3 )
% 0.19/0.45 = X3 ) )
% 0.19/0.45 => ( U
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.one_unique
% 0.19/0.45 thf(fact_74_add_Ol__inv__ex,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ? [X3: a] :
% 0.19/0.45 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 & ( ( add_a_b @ r @ X3 @ X )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.l_inv_ex
% 0.19/0.45 thf(fact_75_add_Oinv__comm,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( ( add_a_b @ r @ X @ Y )
% 0.19/0.45 = ( zero_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ Y @ X )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_comm
% 0.19/0.45 thf(fact_76_r__minus,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 0.19/0.45 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_minus
% 0.19/0.45 thf(fact_77_l__minus,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 0.19/0.45 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_minus
% 0.19/0.45 thf(fact_78_r__neg2,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 0.19/0.45 = Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_neg2
% 0.19/0.45 thf(fact_79_r__neg1,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.45 = Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_neg1
% 0.19/0.45 thf(fact_80_local_Ominus__add,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.45 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.minus_add
% 0.19/0.45 thf(fact_81_add_Oinv__solve__right_H,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 0.19/0.45 = A )
% 0.19/0.45 = ( B
% 0.19/0.45 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_solve_right'
% 0.19/0.45 thf(fact_82_add_Oinv__solve__right,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( A
% 0.19/0.45 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 0.19/0.45 = ( B
% 0.19/0.45 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_solve_right
% 0.19/0.45 thf(fact_83_add_Oinv__solve__left_H,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 0.19/0.45 = A )
% 0.19/0.45 = ( C
% 0.19/0.45 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_solve_left'
% 0.19/0.45 thf(fact_84_add_Oinv__solve__left,axiom,
% 0.19/0.45 ! [A: a,B: a,C: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( A
% 0.19/0.45 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 0.19/0.45 = ( C
% 0.19/0.45 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_solve_left
% 0.19/0.45 thf(fact_85_add_Oinv__mult__group,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.45 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_mult_group
% 0.19/0.45 thf(fact_86_a__transpose__inv,axiom,
% 0.19/0.45 ! [X: a,Y: a,Z: a] :
% 0.19/0.45 ( ( ( add_a_b @ r @ X @ Y )
% 0.19/0.45 = Z )
% 0.19/0.45 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 0.19/0.45 = Y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % a_transpose_inv
% 0.19/0.45 thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
% 0.19/0.45 locali1648887798ac_a_b @ r @ s ).
% 0.19/0.45
% 0.19/0.45 % eq_obj_rng_of_frac_axioms
% 0.19/0.45 thf(fact_88_sum__zero__eq__neg,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ X @ Y )
% 0.19/0.45 = ( zero_a_b @ r ) )
% 0.19/0.45 => ( X
% 0.19/0.45 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_zero_eq_neg
% 0.19/0.45 thf(fact_89_r__neg,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_neg
% 0.19/0.45 thf(fact_90_minus__equality,axiom,
% 0.19/0.45 ! [Y: a,X: a] :
% 0.19/0.45 ( ( ( add_a_b @ r @ Y @ X )
% 0.19/0.45 = ( zero_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_inv_a_b @ r @ X )
% 0.19/0.45 = Y ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_equality
% 0.19/0.45 thf(fact_91_l__neg,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_neg
% 0.19/0.45 thf(fact_92_inv__add,axiom,
% 0.19/0.45 ! [A: a,B: a] :
% 0.19/0.45 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 0.19/0.45 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % inv_add
% 0.19/0.45 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.19/0.45 ~ ! [T: a] :
% 0.19/0.45 ( ( member_a @ T @ s )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ T @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.19/0.45 != ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>\<And>thesis. (\<And>t. \<lbrakk>t \<in> S; t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.19/0.45 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.19/0.45 ~ ! [T2: a] :
% 0.19/0.45 ( ( member_a @ T2 @ s )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ T2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.19/0.45 != ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>\<And>thesis. (\<And>t'. \<lbrakk>t' \<in> S; t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.19/0.45 thf(fact_95_zero__closed,axiom,
% 0.19/0.45 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.45
% 0.19/0.45 % zero_closed
% 0.19/0.45 thf(fact_96_local_Ominus__minus,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 0.19/0.45 = X ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.minus_minus
% 0.19/0.45 thf(fact_97_add_Oinv__closed,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.inv_closed
% 0.19/0.45 thf(fact_98_local_Ominus__zero,axiom,
% 0.19/0.45 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 0.19/0.45 = ( zero_a_b @ r ) ) ).
% 0.19/0.45
% 0.19/0.45 % local.minus_zero
% 0.19/0.45 thf(fact_99_m__closed,axiom,
% 0.19/0.45 ! [X: a,Y: a] :
% 0.19/0.45 ( ( member_a @ X @ s )
% 0.19/0.45 => ( ( member_a @ Y @ s )
% 0.19/0.45 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % m_closed
% 0.19/0.45 thf(fact_100_r__null,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_null
% 0.19/0.45 thf(fact_101_l__null,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_null
% 0.19/0.45 thf(fact_102_r__zero,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.19/0.45 = X ) ) ).
% 0.19/0.45
% 0.19/0.45 % r_zero
% 0.19/0.45 thf(fact_103_l__zero,axiom,
% 0.19/0.45 ! [X: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.19/0.45 = X ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_zero
% 0.19/0.45 thf(fact_104_add_Or__cancel__one_H,axiom,
% 0.19/0.45 ! [X: a,A: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( X
% 0.19/0.45 = ( add_a_b @ r @ A @ X ) )
% 0.19/0.45 = ( A
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.r_cancel_one'
% 0.19/0.45 thf(fact_105_add_Or__cancel__one,axiom,
% 0.19/0.45 ! [X: a,A: a] :
% 0.19/0.45 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.45 => ( ( ( add_a_b @ r @ A @ X )
% 0.19/0.45 = X )
% 0.19/0.45 = ( A
% 0.19/0.45 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.r_cancel_one
% 0.19/0.46 thf(fact_106_add_Ol__cancel__one_H,axiom,
% 0.19/0.46 ! [X: a,A: a] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( X
% 0.19/0.46 = ( add_a_b @ r @ X @ A ) )
% 0.19/0.46 = ( A
% 0.19/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.l_cancel_one'
% 0.19/0.46 thf(fact_107_add_Ol__cancel__one,axiom,
% 0.19/0.46 ! [X: a,A: a] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( ( add_a_b @ r @ X @ A )
% 0.19/0.46 = X )
% 0.19/0.46 = ( A
% 0.19/0.46 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.l_cancel_one
% 0.19/0.46 thf(fact_108_add_Oinv__eq__1__iff,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( ( a_inv_a_b @ r @ X )
% 0.19/0.46 = ( zero_a_b @ r ) )
% 0.19/0.46 = ( X
% 0.19/0.46 = ( zero_a_b @ r ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.inv_eq_1_iff
% 0.19/0.46 thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
% 0.19/0.46 locali880295127ng_a_b @ r @ s ).
% 0.19/0.46
% 0.19/0.46 % mult_submonoid_of_rng_axioms
% 0.19/0.46 thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
% 0.19/0.46 locali807230110ng_a_b @ r @ s ).
% 0.19/0.46
% 0.19/0.46 % mult_submonoid_of_crng_axioms
% 0.19/0.46 thf(fact_111_prod__eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: product_prod_a_a,Z2: product_prod_a_a] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [S: product_prod_a_a,T3: product_prod_a_a] :
% 0.19/0.46 ( ( ( product_fst_a_a @ S )
% 0.19/0.46 = ( product_fst_a_a @ T3 ) )
% 0.19/0.46 & ( ( product_snd_a_a @ S )
% 0.19/0.46 = ( product_snd_a_a @ T3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % prod_eq_iff
% 0.19/0.46 thf(fact_112_submonoid__axioms,axiom,
% 0.19/0.46 locali425460998xt_a_b @ r @ s ).
% 0.19/0.46
% 0.19/0.46 % submonoid_axioms
% 0.19/0.46 thf(fact_113_add_Oint__pow__inv,axiom,
% 0.19/0.46 ! [X: a,I: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.19/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_inv
% 0.19/0.46 thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(11)
% 0.19/0.46 thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(6)
% 0.19/0.46 thf(fact_116_semiring_Ol__null,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
% 0.19/0.46 = ( zero_a_b @ R ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.l_null
% 0.19/0.46 thf(fact_117_semiring_Or__null,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
% 0.19/0.46 = ( zero_a_b @ R ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.r_null
% 0.19/0.46 thf(fact_118_semiring_Or__distr,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
% 0.19/0.46 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.r_distr
% 0.19/0.46 thf(fact_119_add__pow__ldistr__int,axiom,
% 0.19/0.46 ! [A: a,B: a,K: int] :
% 0.19/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 0.19/0.46 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_pow_ldistr_int
% 0.19/0.46 thf(fact_120_add__pow__rdistr__int,axiom,
% 0.19/0.46 ! [A: a,B: a,K: int] :
% 0.19/0.46 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 0.19/0.46 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_pow_rdistr_int
% 0.19/0.46 thf(fact_121_add_Oint__pow__distrib,axiom,
% 0.19/0.46 ! [X: a,Y: a,I: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_distrib
% 0.19/0.46 thf(fact_122_add_Oint__pow__mult__distrib,axiom,
% 0.19/0.46 ! [X: a,Y: a,I: int] :
% 0.19/0.46 ( ( ( add_a_b @ r @ X @ Y )
% 0.19/0.46 = ( add_a_b @ r @ Y @ X ) )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.19/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_mult_distrib
% 0.19/0.46 thf(fact_123_add_Oint__pow__closed,axiom,
% 0.19/0.46 ! [X: a,I: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_closed
% 0.19/0.46 thf(fact_124_add_Oint__pow__one,axiom,
% 0.19/0.46 ! [Z: int] :
% 0.19/0.46 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 0.19/0.46 = ( zero_a_b @ r ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_one
% 0.19/0.46 thf(fact_125_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
% 0.19/0.46 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 0.19/0.46 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.19/0.46 => ( locali425460998xt_a_b @ M @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Localization_Mirabelle_afvchqjmgj.submonoid.is_submonoid
% 0.19/0.46 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.19/0.46 ( ( locali807230110ng_a_b @ R @ S2 )
% 0.19/0.46 => ( locali880295127ng_a_b @ R @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult_submonoid_of_crng.axioms(2)
% 0.19/0.46 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.19/0.46 ( ( locali880295127ng_a_b @ R @ S2 )
% 0.19/0.46 => ( locali425460998xt_a_b @ R @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult_submonoid_of_rng.axioms(2)
% 0.19/0.46 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.19/0.46 ( ( locali1648887798ac_a_b @ R @ S2 )
% 0.19/0.46 => ( locali807230110ng_a_b @ R @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_obj_rng_of_frac.axioms(2)
% 0.19/0.46 thf(fact_129_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 0.19/0.46 ! [M: partia1833973666xt_a_b,S2: set_a,X: a,Y: a] :
% 0.19/0.46 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.19/0.46 => ( ( member_a @ X @ S2 )
% 0.19/0.46 => ( ( member_a @ Y @ S2 )
% 0.19/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
% 0.19/0.46 thf(fact_130_a__minus__def,axiom,
% 0.19/0.46 ( a_minus_a_b
% 0.19/0.46 = ( ^ [R2: partia1833973666xt_a_b,X2: a,Y5: a] : ( add_a_b @ R2 @ X2 @ ( a_inv_a_b @ R2 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % a_minus_def
% 0.19/0.46 thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
% 0.19/0.46 = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(8)
% 0.19/0.46 thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(3)
% 0.19/0.46 thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
% 0.19/0.46 = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(12)
% 0.19/0.46 thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( add_a_b @ R @ X @ Y )
% 0.19/0.46 = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(7)
% 0.19/0.46 thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 0.19/0.46 = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(5)
% 0.19/0.46 thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(1)
% 0.19/0.46 thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( member_a @ ( zero_a_b @ R ) @ ( partia1066395285xt_a_b @ R ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.semiring_simprules(2)
% 0.19/0.46 thf(fact_138_semiring_Ol__distr,axiom,
% 0.19/0.46 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.19/0.46 ( ( semiring_a_b @ R )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.19/0.46 => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 0.19/0.46 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % semiring.l_distr
% 0.19/0.46 thf(fact_139_add_Oint__pow__diff,axiom,
% 0.19/0.46 ! [X: a,N: int,M2: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M2 ) @ X )
% 0.19/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M2 @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_diff
% 0.19/0.46 thf(fact_140_add_Oint__pow__neg,axiom,
% 0.19/0.46 ! [X: a,I: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 0.19/0.46 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_neg
% 0.19/0.46 thf(fact_141_add_Oint__pow__mult,axiom,
% 0.19/0.46 ! [X: a,I: int,J: int] :
% 0.19/0.46 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 0.19/0.46 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.int_pow_mult
% 0.19/0.46 thf(fact_142_subset,axiom,
% 0.19/0.46 ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% 0.19/0.46
% 0.19/0.46 % subset
% 0.19/0.46 thf(fact_143_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
% 0.19/0.46 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 0.19/0.46 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.19/0.46 => ( ord_less_eq_set_a @ S2 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Localization_Mirabelle_afvchqjmgj.submonoid.subset
% 0.19/0.46 thf(fact_144_diff__minus__eq__add,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.19/0.46 = ( plus_plus_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_minus_eq_add
% 0.19/0.46 thf(fact_145_uminus__add__conv__diff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.19/0.46 = ( minus_minus_int @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_add_conv_diff
% 0.19/0.46 thf(fact_146_le__add__diff__inverse,axiom,
% 0.19/0.46 ! [B: int,A: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.19/0.46 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_diff_inverse
% 0.19/0.46 thf(fact_147_le__add__diff__inverse2,axiom,
% 0.19/0.46 ! [B: int,A: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.19/0.46 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_diff_inverse2
% 0.19/0.46 thf(fact_148_add__right__cancel,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.19/0.46 = ( plus_plus_int @ C @ A ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_cancel
% 0.19/0.46 thf(fact_149_add__left__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.19/0.46 = ( plus_plus_int @ A @ C ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_cancel
% 0.19/0.46 thf(fact_150_neg__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ( uminus_uminus_int @ A )
% 0.19/0.46 = ( uminus_uminus_int @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_iff_equal
% 0.19/0.46 thf(fact_151_add_Oinverse__inverse,axiom,
% 0.19/0.46 ! [A: int] :
% 0.19/0.46 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_inverse
% 0.19/0.46 thf(fact_152_add__le__cancel__right,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.19/0.46 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_right
% 0.19/0.46 thf(fact_153_add__le__cancel__left,axiom,
% 0.19/0.46 ! [C: int,A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_left
% 0.19/0.46 thf(fact_154_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: int,A: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.19/0.46 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_155_add__diff__cancel__right_H,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right'
% 0.19/0.46 thf(fact_156_add__diff__cancel__right,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.19/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right
% 0.19/0.46 thf(fact_157_add__diff__cancel__left_H,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left'
% 0.19/0.46 thf(fact_158_add__diff__cancel__left,axiom,
% 0.19/0.46 ! [C: int,A: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.19/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left
% 0.19/0.46 thf(fact_159_diff__add__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_cancel
% 0.19/0.46 thf(fact_160_add__diff__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel
% 0.19/0.46 thf(fact_161_minus__add__distrib,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.19/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_distrib
% 0.19/0.46 thf(fact_162_minus__add__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_cancel
% 0.19/0.46 thf(fact_163_add__minus__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % add_minus_cancel
% 0.19/0.46 thf(fact_164_minus__diff__eq,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.19/0.46 = ( minus_minus_int @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_diff_eq
% 0.19/0.46 thf(fact_165_add__right__imp__eq,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.19/0.46 = ( plus_plus_int @ C @ A ) )
% 0.19/0.46 => ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_imp_eq
% 0.19/0.46 thf(fact_166_add__left__imp__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.19/0.46 = ( plus_plus_int @ A @ C ) )
% 0.19/0.46 => ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_imp_eq
% 0.19/0.46 thf(fact_167_add_Oleft__commute,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.19/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_commute
% 0.19/0.46 thf(fact_168_add_Ocommute,axiom,
% 0.19/0.46 ( plus_plus_int
% 0.19/0.46 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.commute
% 0.19/0.46 thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.19/0.46 = ( plus_plus_int @ C @ A ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_add_class.add.right_cancel
% 0.19/0.46 thf(fact_170_add_Oleft__cancel,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.19/0.46 = ( plus_plus_int @ A @ C ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_cancel
% 0.19/0.46 thf(fact_171_add_Oassoc,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.19/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.assoc
% 0.19/0.46 thf(fact_172_group__cancel_Oadd2,axiom,
% 0.19/0.46 ! [B3: int,K: int,B: int,A: int] :
% 0.19/0.46 ( ( B3
% 0.19/0.46 = ( plus_plus_int @ K @ B ) )
% 0.19/0.46 => ( ( plus_plus_int @ A @ B3 )
% 0.19/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add2
% 0.19/0.46 thf(fact_173_group__cancel_Oadd1,axiom,
% 0.19/0.46 ! [A2: int,K: int,A: int,B: int] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( plus_plus_int @ K @ A ) )
% 0.19/0.46 => ( ( plus_plus_int @ A2 @ B )
% 0.19/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add1
% 0.19/0.46 thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ( plus_plus_int @ I @ K )
% 0.19/0.46 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(4)
% 0.19/0.46 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.19/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.46 thf(fact_176_diff__right__commute,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.19/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_right_commute
% 0.19/0.46 thf(fact_177_diff__eq__diff__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.19/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.19/0.46 = ( minus_minus_int @ C @ D ) )
% 0.19/0.46 => ( ( A = B )
% 0.19/0.46 = ( C = D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_diff_eq
% 0.19/0.46 thf(fact_178_minus__equation__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ( uminus_uminus_int @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( ( uminus_uminus_int @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_equation_iff
% 0.19/0.46 thf(fact_179_equation__minus__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_int @ B ) )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equation_minus_iff
% 0.19/0.46 thf(fact_180_add__le__imp__le__right,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.19/0.46 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_right
% 0.19/0.46 thf(fact_181_add__le__imp__le__left,axiom,
% 0.19/0.46 ! [C: int,A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.19/0.46 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_left
% 0.19/0.46 thf(fact_182_add__right__mono,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_mono
% 0.19/0.46 thf(fact_183_add__left__mono,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_mono
% 0.19/0.46 thf(fact_184_add__mono,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ C @ D )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono
% 0.19/0.46 thf(fact_185_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.19/0.46 ( ( ( ord_less_eq_int @ I @ J )
% 0.19/0.46 & ( ord_less_eq_int @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(1)
% 0.19/0.46 thf(fact_186_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_less_eq_int @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(2)
% 0.19/0.46 thf(fact_187_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.19/0.46 ( ( ( ord_less_eq_int @ I @ J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(3)
% 0.19/0.46 thf(fact_188_diff__eq__diff__less__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.19/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.19/0.46 = ( minus_minus_int @ C @ D ) )
% 0.19/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_diff_less_eq
% 0.19/0.46 thf(fact_189_diff__right__mono,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_right_mono
% 0.19/0.46 thf(fact_190_diff__left__mono,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.19/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_left_mono
% 0.19/0.46 thf(fact_191_diff__mono,axiom,
% 0.19/0.46 ! [A: int,B: int,D: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ D @ C )
% 0.19/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_mono
% 0.19/0.46 thf(fact_192_le__imp__neg__le,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_neg_le
% 0.19/0.46 thf(fact_193_minus__le__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_le_iff
% 0.19/0.46 thf(fact_194_le__minus__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.19/0.46 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_minus_iff
% 0.19/0.46 thf(fact_195_add__implies__diff,axiom,
% 0.19/0.46 ! [C: int,B: int,A: int] :
% 0.19/0.46 ( ( ( plus_plus_int @ C @ B )
% 0.19/0.46 = A )
% 0.19/0.46 => ( C
% 0.19/0.46 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_implies_diff
% 0.19/0.46 thf(fact_196_diff__diff__add,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.19/0.46 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_add
% 0.19/0.46 thf(fact_197_diff__add__eq__diff__diff__swap,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.19/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_eq_diff_diff_swap
% 0.19/0.46 thf(fact_198_diff__add__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.19/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_eq
% 0.19/0.46 thf(fact_199_diff__diff__eq2,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.19/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_eq2
% 0.19/0.46 thf(fact_200_add__diff__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.19/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_eq
% 0.19/0.46 thf(fact_201_eq__diff__eq,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( minus_minus_int @ C @ B ) )
% 0.19/0.46 = ( ( plus_plus_int @ A @ B )
% 0.19/0.46 = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_diff_eq
% 0.19/0.46 thf(fact_202_diff__eq__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.19/0.46 = C )
% 0.19/0.46 = ( A
% 0.19/0.46 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_eq
% 0.19/0.46 thf(fact_203_group__cancel_Osub1,axiom,
% 0.19/0.46 ! [A2: int,K: int,A: int,B: int] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( plus_plus_int @ K @ A ) )
% 0.19/0.46 => ( ( minus_minus_int @ A2 @ B )
% 0.19/0.46 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.sub1
% 0.19/0.46 thf(fact_204_add_Oinverse__distrib__swap,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.19/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_distrib_swap
% 0.19/0.46 thf(fact_205_group__cancel_Oneg1,axiom,
% 0.19/0.46 ! [A2: int,K: int,A: int] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( plus_plus_int @ K @ A ) )
% 0.19/0.46 => ( ( uminus_uminus_int @ A2 )
% 0.19/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.neg1
% 0.19/0.46 thf(fact_206_minus__diff__commute,axiom,
% 0.19/0.46 ! [B: int,A: int] :
% 0.19/0.46 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.19/0.46 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_diff_commute
% 0.19/0.46 thf(fact_207_add__le__add__imp__diff__le,axiom,
% 0.19/0.46 ! [I: int,K: int,N: int,J: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.19/0.46 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.19/0.46 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.19/0.46 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_add_imp_diff_le
% 0.19/0.46 thf(fact_208_add__le__imp__le__diff,axiom,
% 0.19/0.46 ! [I: int,K: int,N: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.19/0.46 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_diff
% 0.19/0.46 thf(fact_209_le__diff__eq,axiom,
% 0.19/0.46 ! [A: int,C: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_eq
% 0.19/0.46 thf(fact_210_diff__le__eq,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.19/0.46 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_eq
% 0.19/0.46 thf(fact_211_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.19/0.46 ( minus_minus_int
% 0.19/0.46 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.19/0.46 thf(fact_212_diff__conv__add__uminus,axiom,
% 0.19/0.46 ( minus_minus_int
% 0.19/0.46 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_conv_add_uminus
% 0.19/0.46 thf(fact_213_group__cancel_Osub2,axiom,
% 0.19/0.46 ! [B3: int,K: int,B: int,A: int] :
% 0.19/0.46 ( ( B3
% 0.19/0.46 = ( plus_plus_int @ K @ B ) )
% 0.19/0.46 => ( ( minus_minus_int @ A @ B3 )
% 0.19/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.sub2
% 0.19/0.46 thf(fact_214_add_Oone__in__subset,axiom,
% 0.19/0.46 ! [H: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( H != bot_bot_set_a )
% 0.19/0.46 => ( ! [X3: a] :
% 0.19/0.46 ( ( member_a @ X3 @ H )
% 0.19/0.46 => ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
% 0.19/0.46 => ( ! [X3: a] :
% 0.19/0.46 ( ( member_a @ X3 @ H )
% 0.19/0.46 => ! [Xa: a] :
% 0.19/0.46 ( ( member_a @ Xa @ H )
% 0.19/0.46 => ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H ) ) )
% 0.19/0.46 => ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.one_in_subset
% 0.19/0.46 thf(fact_215_a__lcos__mult__one,axiom,
% 0.19/0.46 ! [M: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % a_lcos_mult_one
% 0.19/0.46 thf(fact_216_a__lcos__m__assoc,axiom,
% 0.19/0.46 ! [M: set_a,G: a,H2: a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ G @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ H2 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M ) )
% 0.19/0.46 = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % a_lcos_m_assoc
% 0.19/0.46 thf(fact_217_carrier__not__empty,axiom,
% 0.19/0.46 ( ( partia1066395285xt_a_b @ r )
% 0.19/0.46 != bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % carrier_not_empty
% 0.19/0.46 thf(fact_218_a__l__coset__subset__G,axiom,
% 0.19/0.46 ! [H: set_a,X: a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % a_l_coset_subset_G
% 0.19/0.46 thf(fact_219_setadd__subset__G,axiom,
% 0.19/0.46 ! [H: set_a,K2: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % setadd_subset_G
% 0.19/0.46 thf(fact_220_set__add__closed,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ( ord_less_eq_set_a @ B3 @ ( partia1066395285xt_a_b @ r ) )
% 0.19/0.46 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B3 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % set_add_closed
% 0.19/0.46 thf(fact_221_add__additive__subgroups,axiom,
% 0.19/0.46 ! [H: set_a,K2: set_a] :
% 0.19/0.46 ( ( additi2104487374up_a_b @ H @ r )
% 0.19/0.46 => ( ( additi2104487374up_a_b @ K2 @ r )
% 0.19/0.46 => ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_additive_subgroups
% 0.19/0.46 thf(fact_222_Diff__eq__empty__iff,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( ( minus_minus_set_a @ A2 @ B3 )
% 0.19/0.46 = bot_bot_set_a )
% 0.19/0.46 = ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_eq_empty_iff
% 0.19/0.46 thf(fact_223_empty__iff,axiom,
% 0.19/0.46 ! [C: a] :
% 0.19/0.46 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % empty_iff
% 0.19/0.46 thf(fact_224_all__not__in__conv,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( ! [X2: a] :
% 0.19/0.46 ~ ( member_a @ X2 @ A2 ) )
% 0.19/0.46 = ( A2 = bot_bot_set_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % all_not_in_conv
% 0.19/0.46 thf(fact_225_Collect__empty__eq,axiom,
% 0.19/0.46 ! [P: a > $o] :
% 0.19/0.46 ( ( ( collect_a @ P )
% 0.19/0.46 = bot_bot_set_a )
% 0.19/0.46 = ( ! [X2: a] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_empty_eq
% 0.19/0.46 thf(fact_226_empty__Collect__eq,axiom,
% 0.19/0.46 ! [P: a > $o] :
% 0.19/0.46 ( ( bot_bot_set_a
% 0.19/0.46 = ( collect_a @ P ) )
% 0.19/0.46 = ( ! [X2: a] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Collect_eq
% 0.19/0.46 thf(fact_227_subsetI,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ! [X3: a] :
% 0.19/0.46 ( ( member_a @ X3 @ A2 )
% 0.19/0.46 => ( member_a @ X3 @ B3 ) )
% 0.19/0.46 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetI
% 0.19/0.46 thf(fact_228_subset__antisym,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.19/0.46 => ( A2 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_antisym
% 0.19/0.46 thf(fact_229_Compl__subset__Compl__iff,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
% 0.19/0.46 = ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_subset_Compl_iff
% 0.19/0.46 thf(fact_230_Compl__anti__mono,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.19/0.46 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_anti_mono
% 0.19/0.46 thf(fact_231_subset__empty,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.19/0.46 = ( A2 = bot_bot_set_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_empty
% 0.19/0.46 thf(fact_232_empty__subsetI,axiom,
% 0.19/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.19/0.46
% 0.19/0.46 % empty_subsetI
% 0.19/0.46 thf(fact_233_Diff__empty,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_empty
% 0.19/0.46 thf(fact_234_empty__Diff,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
% 0.19/0.46 = bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Diff
% 0.19/0.46 thf(fact_235_Diff__cancel,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( minus_minus_set_a @ A2 @ A2 )
% 0.19/0.46 = bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_cancel
% 0.19/0.46 thf(fact_236_emptyE,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % emptyE
% 0.19/0.46 thf(fact_237_equals0D,axiom,
% 0.19/0.46 ! [A2: set_a,A: a] :
% 0.19/0.46 ( ( A2 = bot_bot_set_a )
% 0.19/0.46 => ~ ( member_a @ A @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % equals0D
% 0.19/0.46 thf(fact_238_equals0I,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ! [Y2: a] :
% 0.19/0.46 ~ ( member_a @ Y2 @ A2 )
% 0.19/0.46 => ( A2 = bot_bot_set_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % equals0I
% 0.19/0.46 thf(fact_239_ex__in__conv,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
% 0.19/0.46 = ( A2 != bot_bot_set_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_in_conv
% 0.19/0.46 thf(fact_240_in__mono,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a,X: a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.19/0.46 => ( ( member_a @ X @ A2 )
% 0.19/0.46 => ( member_a @ X @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_mono
% 0.19/0.46 thf(fact_241_subsetD,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.19/0.46 => ( ( member_a @ C @ A2 )
% 0.19/0.46 => ( member_a @ C @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetD
% 0.19/0.46 thf(fact_242_equalityE,axiom,
% 0.19/0.46 ! [A2: set_a,B3: set_a] :
% 0.19/0.46 ( ( A2 = B3 )
% 0.19/0.46 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.19/0.46 => ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equalityE
% 0.19/0.46 thf(fact_243_subset__eq,axiom,
% 0.19/0.46 ( ord_less_eq_set_a
% 0.19/0.46 = ( ^ [A4: set_a,B4: set_a] :
% 0.19/0.46 ! [X2: a] :
% 0.19/0.46 ( ( member_a @ X2 @ A4 )
% 0.19/0.46 => ( member_a @ X2 @ B4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_eq
% 0.19/0.46
% 0.19/0.46 % Conjectures (1)
% 0.19/0.46 thf(conj_0,conjecture,
% 0.19/0.46 ( ( 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 @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.19/0.46 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 2.53/1.45 ---- Embedded in THF ---
% 2.53/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.53/1.45 %%% Generated on Tue Feb 24 08:26:17 EST 2026
% 2.53/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.53/1.45 %%% Logic specification used:
% 2.53/1.45 %%% thf(spec, logic, $$dhol).
% 2.53/1.45
% 2.53/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2DTF24206.p
% See solution above
% 2.53/1.45 ------------------------
% 2.53/1.48 ---- Cleaned THF ---
% 2.53/1.48 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,
% 2.53/1.48 partia1833973666xt_a_b: $tType ).
% 2.53/1.48
% 2.53/1.48 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.53/1.48 product_prod_a_a: $tType ).
% 2.53/1.48
% 2.53/1.48 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.53/1.48 set_a: $tType ).
% 2.53/1.48
% 2.53/1.48 thf(ty_n_t__Int__Oint,type,
% 2.53/1.48 int: $tType ).
% 2.53/1.48
% 2.53/1.48 thf(ty_n_tf__a,type,
% 2.53/1.48 a: $tType ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
% 2.53/1.48 a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
% 2.53/1.48 additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
% 2.53/1.48 set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
% 2.53/1.48
% 2.53/1.48 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,
% 2.53/1.48 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 2.53/1.48 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 2.53/1.48 minus_minus_int: int > int > int ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 2.53/1.48 minus_minus_set_a: set_a > set_a > set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 2.53/1.48 plus_plus_int: int > int > int ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 2.53/1.48 uminus_uminus_int: int > int ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 2.53/1.48 uminus_uminus_set_a: set_a > set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oeq__obj__rng__of__frac_001tf__a_001tf__b,type,
% 2.53/1.48 locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__crng_001tf__a_001tf__b,type,
% 2.53/1.48 locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__rng_001tf__a_001tf__b,type,
% 2.53/1.48 locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Osubmonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 2.53/1.48 locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 2.53/1.48 bot_bot_set_a: set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.53/1.48 ord_less_eq_int: int > int > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.53/1.48 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
% 2.53/1.48 product_fst_a_a: product_prod_a_a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
% 2.53/1.48 product_snd_a_a: product_prod_a_a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 2.53/1.48 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 2.53/1.48 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 2.53/1.48 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 2.53/1.48 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 2.53/1.48 zero_a_b: partia1833973666xt_a_b > a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 2.53/1.48 semiring_a_b: partia1833973666xt_a_b > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.53/1.48 collect_a: ( a > $o ) > set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_c_member_001tf__a,type,
% 2.53/1.48 member_a: a > set_a > $o ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_R,type,
% 2.53/1.48 r: partia1833973666xt_a_b ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_S,type,
% 2.53/1.48 s: set_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_r,type,
% 2.53/1.48 r2: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_r_H,type,
% 2.53/1.48 r3: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_s,type,
% 2.53/1.48 s2: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_s_H,type,
% 2.53/1.48 s3: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_t_H____,type,
% 2.53/1.48 t: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_t____,type,
% 2.53/1.48 t2: a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_x_H____,type,
% 2.53/1.48 x: product_prod_a_a ).
% 2.53/1.48
% 2.53/1.48 thf(sy_v_y_H____,type,
% 2.53/1.48 y: product_prod_a_a ).
% 2.53/1.48
% 2.53/1.48 thf(fact_0_f18,axiom,
% 2.53/1.48 member_a @ t @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_1_f17,axiom,
% 2.53/1.48 member_a @ t2 @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_2_right__add__eq,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( A = B )
% 2.53/1.48 => ( ( add_a_b @ r @ C @ A )
% 2.53/1.48 = ( add_a_b @ r @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_3_local_Oright__minus__eq,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( A = B )
% 2.53/1.48 => ( ( a_minus_a_b @ r @ C @ A )
% 2.53/1.48 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_4_m__assoc,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_5_m__comm,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_6_m__lcomm,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_7_add_Om__assoc,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 2.53/1.48 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_8_add_Om__comm,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ X @ Y )
% 2.53/1.48 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_9_add_Om__lcomm,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 2.53/1.48 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_10_f20,axiom,
% 2.53/1.48 member_a @ s3 @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_11_f19,axiom,
% 2.53/1.48 member_a @ s2 @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_12_f41,axiom,
% 2.53/1.48 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_13_f42,axiom,
% 2.53/1.48 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_14_l__distr,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_15_r__distr,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_16_four__elem__comm,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a,D: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_17_right__inv__add,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 2.53/1.48 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_18_f21,axiom,
% 2.53/1.48 member_a @ ( product_snd_a_a @ y ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_19_f40,axiom,
% 2.53/1.48 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_20_f22,axiom,
% 2.53/1.48 member_a @ ( product_fst_a_a @ x ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 2.53/1.48 ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_22_f15,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_24_f28,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_25_f11,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_26_f10,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_27_f27,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_28_f35,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_31_f16,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_32_f24,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_33_f9,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_35_f23,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_36_f34,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_37_f36,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_38_f13,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 2.53/1.48 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_40_f12,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_41_f25,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_42_f32,axiom,
% 2.53/1.48 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_43_f26,axiom,
% 2.53/1.48 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_44_f29,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_45_mem__Collect__eq,axiom,
% 2.53/1.48 ! [A: a,P: a > $o] :
% 2.53/1.48 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.53/1.48 = ( P @ A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_46_Collect__mem__eq,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( collect_a
% 2.53/1.48 @ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
% 2.53/1.48 = A2 ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_47_Collect__cong,axiom,
% 2.53/1.48 ! [P: a > $o,Q: a > $o] :
% 2.53/1.48 ( ! [X3: a] :
% 2.53/1.48 ( ( P @ X3 )
% 2.53/1.48 = ( Q @ X3 ) )
% 2.53/1.48 => ( ( collect_a @ P )
% 2.53/1.48 = ( collect_a @ Q ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_48_f30,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_49_f33,axiom,
% 2.53/1.48 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_51_f14,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 2.53/1.48 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_53_f31,axiom,
% 2.53/1.48 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 2.53/1.48 = ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_54_f37,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_55_f39,axiom,
% 2.53/1.48 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_56_f38,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_57_f5,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_58_f7,axiom,
% 2.53/1.48 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_59_minus__closed,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_60_add_Om__closed,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_61_local_Oadd_Oright__cancel,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ Y @ X )
% 2.53/1.48 = ( add_a_b @ r @ Z @ X ) )
% 2.53/1.48 = ( Y = Z ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_62_semiring__simprules_I3_J,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_63_f4,axiom,
% 2.53/1.48 member_a @ t2 @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_64_f6,axiom,
% 2.53/1.48 member_a @ t @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_65_f8,axiom,
% 2.53/1.48 member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_66_local_Osemiring__axioms,axiom,
% 2.53/1.48 semiring_a_b @ r ).
% 2.53/1.48
% 2.53/1.48 thf(fact_67_minus__eq,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( a_minus_a_b @ r @ X @ Y )
% 2.53/1.48 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_68_prod__eqI,axiom,
% 2.53/1.48 ! [P2: product_prod_a_a,Q2: product_prod_a_a] :
% 2.53/1.48 ( ( ( product_fst_a_a @ P2 )
% 2.53/1.48 = ( product_fst_a_a @ Q2 ) )
% 2.53/1.48 => ( ( ( product_snd_a_a @ P2 )
% 2.53/1.48 = ( product_snd_a_a @ Q2 ) )
% 2.53/1.48 => ( P2 = Q2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_69_exE__realizer_H,axiom,
% 2.53/1.48 ! [P: a > a > $o,P2: product_prod_a_a] :
% 2.53/1.48 ( ( P @ ( product_snd_a_a @ P2 ) @ ( product_fst_a_a @ P2 ) )
% 2.53/1.48 => ~ ! [X3: a,Y2: a] :
% 2.53/1.48 ~ ( P @ Y2 @ X3 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_70_prod_Oexpand,axiom,
% 2.53/1.48 ! [Prod: product_prod_a_a,Prod2: product_prod_a_a] :
% 2.53/1.48 ( ( ( ( product_fst_a_a @ Prod )
% 2.53/1.48 = ( product_fst_a_a @ Prod2 ) )
% 2.53/1.48 & ( ( product_snd_a_a @ Prod )
% 2.53/1.48 = ( product_snd_a_a @ Prod2 ) ) )
% 2.53/1.48 => ( Prod = Prod2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_71_local_Ominus__unique,axiom,
% 2.53/1.48 ! [Y: a,X: a,Y3: a] :
% 2.53/1.48 ( ( ( add_a_b @ r @ Y @ X )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ X @ Y3 )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( Y = Y3 ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_72_add_Or__inv__ex,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ? [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 & ( ( add_a_b @ r @ X @ X3 )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_73_add_Oone__unique,axiom,
% 2.53/1.48 ! [U: a] :
% 2.53/1.48 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ! [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ U @ X3 )
% 2.53/1.48 = X3 ) )
% 2.53/1.48 => ( U
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_74_add_Ol__inv__ex,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ? [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 & ( ( add_a_b @ r @ X3 @ X )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_75_add_Oinv__comm,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( ( add_a_b @ r @ X @ Y )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ Y @ X )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_76_r__minus,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 2.53/1.48 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_77_l__minus,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 2.53/1.48 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_78_r__neg2,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 2.53/1.48 = Y ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_79_r__neg1,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = Y ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_80_local_Ominus__add,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_81_add_Oinv__solve__right_H,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 2.53/1.48 = A )
% 2.53/1.48 = ( B
% 2.53/1.48 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_82_add_Oinv__solve__right,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( A
% 2.53/1.48 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 2.53/1.48 = ( B
% 2.53/1.48 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_83_add_Oinv__solve__left_H,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 2.53/1.48 = A )
% 2.53/1.48 = ( C
% 2.53/1.48 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_84_add_Oinv__solve__left,axiom,
% 2.53/1.48 ! [A: a,B: a,C: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( A
% 2.53/1.48 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 2.53/1.48 = ( C
% 2.53/1.48 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_85_add_Oinv__mult__group,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_86_a__transpose__inv,axiom,
% 2.53/1.48 ! [X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( ( add_a_b @ r @ X @ Y )
% 2.53/1.48 = Z )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 2.53/1.48 = Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
% 2.53/1.48 locali1648887798ac_a_b @ r @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_88_sum__zero__eq__neg,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ X @ Y )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 => ( X
% 2.53/1.48 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_89_r__neg,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_90_minus__equality,axiom,
% 2.53/1.48 ! [Y: a,X: a] :
% 2.53/1.48 ( ( ( add_a_b @ r @ Y @ X )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_inv_a_b @ r @ X )
% 2.53/1.48 = Y ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_91_l__neg,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_92_inv__add,axiom,
% 2.53/1.48 ! [A: a,B: a] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 2.53/1.48 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.53/1.48 ~ ! [T: a] :
% 2.53/1.48 ( ( member_a @ T @ s )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ T @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 2.53/1.48 != ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.53/1.48 ~ ! [T2: a] :
% 2.53/1.48 ( ( member_a @ T2 @ s )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ T2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 2.53/1.48 != ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_95_zero__closed,axiom,
% 2.53/1.48 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_96_local_Ominus__minus,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 2.53/1.48 = X ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_97_add_Oinv__closed,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_98_local_Ominus__zero,axiom,
% 2.53/1.48 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_99_m__closed,axiom,
% 2.53/1.48 ! [X: a,Y: a] :
% 2.53/1.48 ( ( member_a @ X @ s )
% 2.53/1.48 => ( ( member_a @ Y @ s )
% 2.53/1.48 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_100_r__null,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_101_l__null,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_102_r__zero,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 2.53/1.48 = X ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_103_l__zero,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 2.53/1.48 = X ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_104_add_Or__cancel__one_H,axiom,
% 2.53/1.48 ! [X: a,A: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( X
% 2.53/1.48 = ( add_a_b @ r @ A @ X ) )
% 2.53/1.48 = ( A
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_105_add_Or__cancel__one,axiom,
% 2.53/1.48 ! [X: a,A: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ A @ X )
% 2.53/1.48 = X )
% 2.53/1.48 = ( A
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_106_add_Ol__cancel__one_H,axiom,
% 2.53/1.48 ! [X: a,A: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( X
% 2.53/1.48 = ( add_a_b @ r @ X @ A ) )
% 2.53/1.48 = ( A
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_107_add_Ol__cancel__one,axiom,
% 2.53/1.48 ! [X: a,A: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( add_a_b @ r @ X @ A )
% 2.53/1.48 = X )
% 2.53/1.48 = ( A
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_108_add_Oinv__eq__1__iff,axiom,
% 2.53/1.48 ! [X: a] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ( a_inv_a_b @ r @ X )
% 2.53/1.48 = ( zero_a_b @ r ) )
% 2.53/1.48 = ( X
% 2.53/1.48 = ( zero_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
% 2.53/1.48 locali880295127ng_a_b @ r @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
% 2.53/1.48 locali807230110ng_a_b @ r @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_111_prod__eq__iff,axiom,
% 2.53/1.48 ( ( ^ [Y4: product_prod_a_a,Z2: product_prod_a_a] : ( Y4 = Z2 ) )
% 2.53/1.48 = ( ^ [S: product_prod_a_a,T3: product_prod_a_a] :
% 2.53/1.48 ( ( ( product_fst_a_a @ S )
% 2.53/1.48 = ( product_fst_a_a @ T3 ) )
% 2.53/1.48 & ( ( product_snd_a_a @ S )
% 2.53/1.48 = ( product_snd_a_a @ T3 ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_112_submonoid__axioms,axiom,
% 2.53/1.48 locali425460998xt_a_b @ r @ s ).
% 2.53/1.48
% 2.53/1.48 thf(fact_113_add_Oint__pow__inv,axiom,
% 2.53/1.48 ! [X: a,I: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 2.53/1.48 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
% 2.53/1.48 = X ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
% 2.53/1.48 = X ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_116_semiring_Ol__null,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
% 2.53/1.48 = ( zero_a_b @ R ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_117_semiring_Or__null,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
% 2.53/1.48 = ( zero_a_b @ R ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_118_semiring_Or__distr,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_119_add__pow__ldistr__int,axiom,
% 2.53/1.48 ! [A: a,B: a,K: int] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 2.53/1.48 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_120_add__pow__rdistr__int,axiom,
% 2.53/1.48 ! [A: a,B: a,K: int] :
% 2.53/1.48 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 2.53/1.48 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_121_add_Oint__pow__distrib,axiom,
% 2.53/1.48 ! [X: a,Y: a,I: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_122_add_Oint__pow__mult__distrib,axiom,
% 2.53/1.48 ! [X: a,Y: a,I: int] :
% 2.53/1.48 ( ( ( add_a_b @ r @ X @ Y )
% 2.53/1.48 = ( add_a_b @ r @ Y @ X ) )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 2.53/1.48 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_123_add_Oint__pow__closed,axiom,
% 2.53/1.48 ! [X: a,I: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_124_add_Oint__pow__one,axiom,
% 2.53/1.48 ! [Z: int] :
% 2.53/1.48 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 2.53/1.48 = ( zero_a_b @ r ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_125_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
% 2.53/1.48 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 2.53/1.48 ( ( locali425460998xt_a_b @ M @ S2 )
% 2.53/1.48 => ( locali425460998xt_a_b @ M @ S2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 2.53/1.48 ( ( locali807230110ng_a_b @ R @ S2 )
% 2.53/1.48 => ( locali880295127ng_a_b @ R @ S2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 2.53/1.48 ( ( locali880295127ng_a_b @ R @ S2 )
% 2.53/1.48 => ( locali425460998xt_a_b @ R @ S2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 2.53/1.48 ( ( locali1648887798ac_a_b @ R @ S2 )
% 2.53/1.48 => ( locali807230110ng_a_b @ R @ S2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_129_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 2.53/1.48 ! [M: partia1833973666xt_a_b,S2: set_a,X: a,Y: a] :
% 2.53/1.48 ( ( locali425460998xt_a_b @ M @ S2 )
% 2.53/1.48 => ( ( member_a @ X @ S2 )
% 2.53/1.48 => ( ( member_a @ Y @ S2 )
% 2.53/1.48 => ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S2 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_130_a__minus__def,axiom,
% 2.53/1.48 ( a_minus_a_b
% 2.53/1.48 = ( ^ [R2: partia1833973666xt_a_b,X2: a,Y5: a] : ( add_a_b @ R2 @ X2 @ ( a_inv_a_b @ R2 @ Y5 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
% 2.53/1.48 = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( add_a_b @ R @ X @ Y )
% 2.53/1.48 = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 2.53/1.48 = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( member_a @ ( zero_a_b @ R ) @ ( partia1066395285xt_a_b @ R ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_138_semiring_Ol__distr,axiom,
% 2.53/1.48 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 2.53/1.48 ( ( semiring_a_b @ R )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 2.53/1.48 => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 2.53/1.48 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_139_add_Oint__pow__diff,axiom,
% 2.53/1.48 ! [X: a,N: int,M2: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M2 ) @ X )
% 2.53/1.48 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M2 @ X ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_140_add_Oint__pow__neg,axiom,
% 2.53/1.48 ! [X: a,I: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 2.53/1.48 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_141_add_Oint__pow__mult,axiom,
% 2.53/1.48 ! [X: a,I: int,J: int] :
% 2.53/1.48 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 2.53/1.48 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_142_subset,axiom,
% 2.53/1.48 ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_143_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
% 2.53/1.48 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 2.53/1.48 ( ( locali425460998xt_a_b @ M @ S2 )
% 2.53/1.48 => ( ord_less_eq_set_a @ S2 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_144_diff__minus__eq__add,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 2.53/1.48 = ( plus_plus_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_145_uminus__add__conv__diff,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 2.53/1.48 = ( minus_minus_int @ B @ A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_146_le__add__diff__inverse,axiom,
% 2.53/1.48 ! [B: int,A: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ B @ A )
% 2.53/1.48 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 2.53/1.48 = A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_147_le__add__diff__inverse2,axiom,
% 2.53/1.48 ! [B: int,A: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ B @ A )
% 2.53/1.48 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 2.53/1.48 = A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_148_add__right__cancel,axiom,
% 2.53/1.48 ! [B: int,A: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ B @ A )
% 2.53/1.48 = ( plus_plus_int @ C @ A ) )
% 2.53/1.48 = ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_149_add__left__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ A @ B )
% 2.53/1.48 = ( plus_plus_int @ A @ C ) )
% 2.53/1.48 = ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_150_neg__equal__iff__equal,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( ( uminus_uminus_int @ A )
% 2.53/1.48 = ( uminus_uminus_int @ B ) )
% 2.53/1.48 = ( A = B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_151_add_Oinverse__inverse,axiom,
% 2.53/1.48 ! [A: int] :
% 2.53/1.48 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 2.53/1.48 = A ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_152_add__le__cancel__right,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.53/1.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_153_add__le__cancel__left,axiom,
% 2.53/1.48 ! [C: int,A: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.53/1.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_154_neg__le__iff__le,axiom,
% 2.53/1.48 ! [B: int,A: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 2.53/1.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_155_add__diff__cancel__right_H,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 2.53/1.48 = A ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_156_add__diff__cancel__right,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.53/1.48 = ( minus_minus_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_157_add__diff__cancel__left_H,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 2.53/1.48 = B ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_158_add__diff__cancel__left,axiom,
% 2.53/1.48 ! [C: int,A: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.53/1.48 = ( minus_minus_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_159_diff__add__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 2.53/1.48 = A ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_160_add__diff__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 2.53/1.48 = A ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_161_minus__add__distrib,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 2.53/1.48 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_162_minus__add__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 2.53/1.48 = B ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_163_add__minus__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 2.53/1.48 = B ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_164_minus__diff__eq,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 2.53/1.48 = ( minus_minus_int @ B @ A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_165_add__right__imp__eq,axiom,
% 2.53/1.48 ! [B: int,A: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ B @ A )
% 2.53/1.48 = ( plus_plus_int @ C @ A ) )
% 2.53/1.48 => ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_166_add__left__imp__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ A @ B )
% 2.53/1.48 = ( plus_plus_int @ A @ C ) )
% 2.53/1.48 => ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_167_add_Oleft__commute,axiom,
% 2.53/1.48 ! [B: int,A: int,C: int] :
% 2.53/1.48 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 2.53/1.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_168_add_Ocommute,axiom,
% 2.53/1.48 ( plus_plus_int
% 2.53/1.48 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
% 2.53/1.48 ! [B: int,A: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ B @ A )
% 2.53/1.48 = ( plus_plus_int @ C @ A ) )
% 2.53/1.48 = ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_170_add_Oleft__cancel,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ A @ B )
% 2.53/1.48 = ( plus_plus_int @ A @ C ) )
% 2.53/1.48 = ( B = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_171_add_Oassoc,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 2.53/1.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_172_group__cancel_Oadd2,axiom,
% 2.53/1.48 ! [B3: int,K: int,B: int,A: int] :
% 2.53/1.48 ( ( B3
% 2.53/1.48 = ( plus_plus_int @ K @ B ) )
% 2.53/1.48 => ( ( plus_plus_int @ A @ B3 )
% 2.53/1.48 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_173_group__cancel_Oadd1,axiom,
% 2.53/1.48 ! [A2: int,K: int,A: int,B: int] :
% 2.53/1.48 ( ( A2
% 2.53/1.48 = ( plus_plus_int @ K @ A ) )
% 2.53/1.48 => ( ( plus_plus_int @ A2 @ B )
% 2.53/1.48 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.53/1.48 ! [I: int,J: int,K: int,L: int] :
% 2.53/1.48 ( ( ( I = J )
% 2.53/1.48 & ( K = L ) )
% 2.53/1.48 => ( ( plus_plus_int @ I @ K )
% 2.53/1.48 = ( plus_plus_int @ J @ L ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 2.53/1.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_176_diff__right__commute,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 2.53/1.48 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_177_diff__eq__diff__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int,D: int] :
% 2.53/1.48 ( ( ( minus_minus_int @ A @ B )
% 2.53/1.48 = ( minus_minus_int @ C @ D ) )
% 2.53/1.48 => ( ( A = B )
% 2.53/1.48 = ( C = D ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_178_minus__equation__iff,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( ( uminus_uminus_int @ A )
% 2.53/1.48 = B )
% 2.53/1.48 = ( ( uminus_uminus_int @ B )
% 2.53/1.48 = A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_179_equation__minus__iff,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( A
% 2.53/1.48 = ( uminus_uminus_int @ B ) )
% 2.53/1.48 = ( B
% 2.53/1.48 = ( uminus_uminus_int @ A ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_180_add__le__imp__le__right,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 2.53/1.48 => ( ord_less_eq_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_181_add__le__imp__le__left,axiom,
% 2.53/1.48 ! [C: int,A: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 2.53/1.48 => ( ord_less_eq_int @ A @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_182_add__right__mono,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_183_add__left__mono,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_184_add__mono,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int,D: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ( ord_less_eq_int @ C @ D )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_185_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.53/1.48 ! [I: int,J: int,K: int,L: int] :
% 2.53/1.48 ( ( ( ord_less_eq_int @ I @ J )
% 2.53/1.48 & ( ord_less_eq_int @ K @ L ) )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_186_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.53/1.48 ! [I: int,J: int,K: int,L: int] :
% 2.53/1.48 ( ( ( I = J )
% 2.53/1.48 & ( ord_less_eq_int @ K @ L ) )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_187_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.53/1.48 ! [I: int,J: int,K: int,L: int] :
% 2.53/1.48 ( ( ( ord_less_eq_int @ I @ J )
% 2.53/1.48 & ( K = L ) )
% 2.53/1.48 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_188_diff__eq__diff__less__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int,D: int] :
% 2.53/1.48 ( ( ( minus_minus_int @ A @ B )
% 2.53/1.48 = ( minus_minus_int @ C @ D ) )
% 2.53/1.48 => ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_189_diff__right__mono,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_190_diff__left__mono,axiom,
% 2.53/1.48 ! [B: int,A: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ B @ A )
% 2.53/1.48 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_191_diff__mono,axiom,
% 2.53/1.48 ! [A: int,B: int,D: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ( ord_less_eq_int @ D @ C )
% 2.53/1.48 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_192_le__imp__neg__le,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ B )
% 2.53/1.48 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_193_minus__le__iff,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 2.53/1.48 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_194_le__minus__iff,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 2.53/1.48 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_195_add__implies__diff,axiom,
% 2.53/1.48 ! [C: int,B: int,A: int] :
% 2.53/1.48 ( ( ( plus_plus_int @ C @ B )
% 2.53/1.48 = A )
% 2.53/1.48 => ( C
% 2.53/1.48 = ( minus_minus_int @ A @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_196_diff__diff__add,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 2.53/1.48 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_197_diff__add__eq__diff__diff__swap,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 2.53/1.48 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_198_diff__add__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 2.53/1.48 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_199_diff__diff__eq2,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 2.53/1.48 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_200_add__diff__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 2.53/1.48 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_201_eq__diff__eq,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( A
% 2.53/1.48 = ( minus_minus_int @ C @ B ) )
% 2.53/1.48 = ( ( plus_plus_int @ A @ B )
% 2.53/1.48 = C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_202_diff__eq__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ( minus_minus_int @ A @ B )
% 2.53/1.48 = C )
% 2.53/1.48 = ( A
% 2.53/1.48 = ( plus_plus_int @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_203_group__cancel_Osub1,axiom,
% 2.53/1.48 ! [A2: int,K: int,A: int,B: int] :
% 2.53/1.48 ( ( A2
% 2.53/1.48 = ( plus_plus_int @ K @ A ) )
% 2.53/1.48 => ( ( minus_minus_int @ A2 @ B )
% 2.53/1.48 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_204_add_Oinverse__distrib__swap,axiom,
% 2.53/1.48 ! [A: int,B: int] :
% 2.53/1.48 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 2.53/1.48 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_205_group__cancel_Oneg1,axiom,
% 2.53/1.48 ! [A2: int,K: int,A: int] :
% 2.53/1.48 ( ( A2
% 2.53/1.48 = ( plus_plus_int @ K @ A ) )
% 2.53/1.48 => ( ( uminus_uminus_int @ A2 )
% 2.53/1.48 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_206_minus__diff__commute,axiom,
% 2.53/1.48 ! [B: int,A: int] :
% 2.53/1.48 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 2.53/1.48 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_207_add__le__add__imp__diff__le,axiom,
% 2.53/1.48 ! [I: int,K: int,N: int,J: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 2.53/1.48 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 2.53/1.48 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 2.53/1.48 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 2.53/1.48 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_208_add__le__imp__le__diff,axiom,
% 2.53/1.48 ! [I: int,K: int,N: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 2.53/1.48 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_209_le__diff__eq,axiom,
% 2.53/1.48 ! [A: int,C: int,B: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 2.53/1.48 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_210_diff__le__eq,axiom,
% 2.53/1.48 ! [A: int,B: int,C: int] :
% 2.53/1.48 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 2.53/1.48 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_211_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 2.53/1.48 ( minus_minus_int
% 2.53/1.48 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_212_diff__conv__add__uminus,axiom,
% 2.53/1.48 ( minus_minus_int
% 2.53/1.48 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_213_group__cancel_Osub2,axiom,
% 2.53/1.48 ! [B3: int,K: int,B: int,A: int] :
% 2.53/1.48 ( ( B3
% 2.53/1.48 = ( plus_plus_int @ K @ B ) )
% 2.53/1.48 => ( ( minus_minus_int @ A @ B3 )
% 2.53/1.48 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_214_add_Oone__in__subset,axiom,
% 2.53/1.48 ! [H: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( H != bot_bot_set_a )
% 2.53/1.48 => ( ! [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ H )
% 2.53/1.48 => ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
% 2.53/1.48 => ( ! [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ H )
% 2.53/1.48 => ! [Xa: a] :
% 2.53/1.48 ( ( member_a @ Xa @ H )
% 2.53/1.48 => ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H ) ) )
% 2.53/1.48 => ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_215_a__lcos__mult__one,axiom,
% 2.53/1.48 ! [M: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
% 2.53/1.48 = M ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_216_a__lcos__m__assoc,axiom,
% 2.53/1.48 ! [M: set_a,G: a,H2: a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ G @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ H2 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M ) )
% 2.53/1.48 = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M ) ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_217_carrier__not__empty,axiom,
% 2.53/1.48 ( ( partia1066395285xt_a_b @ r )
% 2.53/1.48 != bot_bot_set_a ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_218_a__l__coset__subset__G,axiom,
% 2.53/1.48 ! [H: set_a,X: a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_219_setadd__subset__G,axiom,
% 2.53/1.48 ! [H: set_a,K2: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_220_set__add__closed,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ( ord_less_eq_set_a @ B3 @ ( partia1066395285xt_a_b @ r ) )
% 2.53/1.48 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B3 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_221_add__additive__subgroups,axiom,
% 2.53/1.48 ! [H: set_a,K2: set_a] :
% 2.53/1.48 ( ( additi2104487374up_a_b @ H @ r )
% 2.53/1.48 => ( ( additi2104487374up_a_b @ K2 @ r )
% 2.53/1.48 => ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_222_Diff__eq__empty__iff,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( ( minus_minus_set_a @ A2 @ B3 )
% 2.53/1.48 = bot_bot_set_a )
% 2.53/1.48 = ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_223_empty__iff,axiom,
% 2.53/1.48 ! [C: a] :
% 2.53/1.48 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_224_all__not__in__conv,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( ! [X2: a] :
% 2.53/1.48 ~ ( member_a @ X2 @ A2 ) )
% 2.53/1.48 = ( A2 = bot_bot_set_a ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_225_Collect__empty__eq,axiom,
% 2.53/1.48 ! [P: a > $o] :
% 2.53/1.48 ( ( ( collect_a @ P )
% 2.53/1.48 = bot_bot_set_a )
% 2.53/1.48 = ( ! [X2: a] :
% 2.53/1.48 ~ ( P @ X2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_226_empty__Collect__eq,axiom,
% 2.53/1.48 ! [P: a > $o] :
% 2.53/1.48 ( ( bot_bot_set_a
% 2.53/1.48 = ( collect_a @ P ) )
% 2.53/1.48 = ( ! [X2: a] :
% 2.53/1.48 ~ ( P @ X2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_227_subsetI,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ! [X3: a] :
% 2.53/1.48 ( ( member_a @ X3 @ A2 )
% 2.53/1.48 => ( member_a @ X3 @ B3 ) )
% 2.53/1.48 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_228_subset__antisym,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 2.53/1.48 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 2.53/1.48 => ( A2 = B3 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_229_Compl__subset__Compl__iff,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
% 2.53/1.48 = ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_230_Compl__anti__mono,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 2.53/1.48 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_231_subset__empty,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 2.53/1.48 = ( A2 = bot_bot_set_a ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_232_empty__subsetI,axiom,
% 2.53/1.48 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_233_Diff__empty,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
% 2.53/1.48 = A2 ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_234_empty__Diff,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
% 2.53/1.48 = bot_bot_set_a ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_235_Diff__cancel,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( minus_minus_set_a @ A2 @ A2 )
% 2.53/1.48 = bot_bot_set_a ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_236_emptyE,axiom,
% 2.53/1.48 ! [A: a] :
% 2.53/1.48 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_237_equals0D,axiom,
% 2.53/1.48 ! [A2: set_a,A: a] :
% 2.53/1.48 ( ( A2 = bot_bot_set_a )
% 2.53/1.48 => ~ ( member_a @ A @ A2 ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_238_equals0I,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ! [Y2: a] :
% 2.53/1.48 ~ ( member_a @ Y2 @ A2 )
% 2.53/1.48 => ( A2 = bot_bot_set_a ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_239_ex__in__conv,axiom,
% 2.53/1.48 ! [A2: set_a] :
% 2.53/1.48 ( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
% 2.53/1.48 = ( A2 != bot_bot_set_a ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_240_in__mono,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a,X: a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 2.53/1.48 => ( ( member_a @ X @ A2 )
% 2.53/1.48 => ( member_a @ X @ B3 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_241_subsetD,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a,C: a] :
% 2.53/1.48 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 2.53/1.48 => ( ( member_a @ C @ A2 )
% 2.53/1.48 => ( member_a @ C @ B3 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_242_equalityE,axiom,
% 2.53/1.48 ! [A2: set_a,B3: set_a] :
% 2.53/1.48 ( ( A2 = B3 )
% 2.53/1.48 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 2.53/1.48 => ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(fact_243_subset__eq,axiom,
% 2.53/1.48 ( ord_less_eq_set_a
% 2.53/1.48 = ( ^ [A4: set_a,B4: set_a] :
% 2.53/1.48 ! [X2: a] :
% 2.53/1.48 ( ( member_a @ X2 @ A4 )
% 2.53/1.48 => ( member_a @ X2 @ B4 ) ) ) ) ).
% 2.53/1.48
% 2.53/1.48 thf(conj_0,conjecture,
% 2.53/1.48 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 2.53/1.48 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 2.53/1.48 ------------------------
% 2.70/1.51 % (24401)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_24206 for (2999ds/27Mi)
% 2.70/1.51 % (24404)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_24206 for (2999ds/275Mi)
% 2.70/1.51 % (24403)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_24206 for (2999ds/2Mi)
% 2.70/1.51 % (24403)Instruction limit reached!
% 2.70/1.51 % (24403)------------------------------
% 2.70/1.51 % (24403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.51 % (24403)Termination reason: Unknown
% 2.70/1.51 % (24403)Termination phase: shuffling
% 2.70/1.51
% 2.70/1.51 % (24403)Memory used [KB]: 1279
% 2.70/1.51 % (24403)Time elapsed: 0.003 s
% 2.70/1.51 % (24403)Instructions burned: 3 (million)
% 2.70/1.51 % (24403)------------------------------
% 2.70/1.51 % (24403)------------------------------
% 2.70/1.51 % (24402)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/2Mi)
% 2.70/1.51 % (24405)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/18Mi)
% 2.70/1.51 % (24402)Instruction limit reached!
% 2.70/1.51 % (24402)------------------------------
% 2.70/1.51 % (24402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.51 % (24402)Termination reason: Unknown
% 2.70/1.51 % (24402)Termination phase: shuffling
% 2.70/1.51
% 2.70/1.51 % (24402)Memory used [KB]: 1151
% 2.70/1.51 % (24402)Time elapsed: 0.003 s
% 2.70/1.51 % (24402)Instructions burned: 2 (million)
% 2.70/1.51 % (24402)------------------------------
% 2.70/1.51 % (24402)------------------------------
% 2.70/1.52 % (24401)Instruction limit reached!
% 2.70/1.52 % (24401)------------------------------
% 2.70/1.52 % (24401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.52 % (24401)Termination reason: Unknown
% 2.70/1.52 % (24401)Termination phase: Property scanning
% 2.70/1.52
% 2.70/1.52 % (24401)Memory used [KB]: 1407
% 2.70/1.52 % (24401)Time elapsed: 0.010 s
% 2.70/1.52 % (24401)Instructions burned: 27 (million)
% 2.70/1.52 % (24401)------------------------------
% 2.70/1.52 % (24401)------------------------------
% 2.70/1.52 % (24399)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/183Mi)
% 2.70/1.52 % (24406)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/3Mi)
% 2.70/1.52 % (24406)Instruction limit reached!
% 2.70/1.52 % (24406)------------------------------
% 2.70/1.52 % (24406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.52 % (24406)Termination reason: Unknown
% 2.70/1.52 % (24406)Termination phase: shuffling
% 2.70/1.52
% 2.70/1.52 % (24406)Memory used [KB]: 1279
% 2.70/1.52 % (24406)Time elapsed: 0.002 s
% 2.70/1.52 % (24406)Instructions burned: 3 (million)
% 2.70/1.52 % (24406)------------------------------
% 2.70/1.52 % (24406)------------------------------
% 2.70/1.52 % (24400)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_24206 for (2999ds/4Mi)
% 2.70/1.52 % (24405)Instruction limit reached!
% 2.70/1.52 % (24405)------------------------------
% 2.70/1.52 % (24405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.52 % (24405)Termination reason: Unknown
% 2.70/1.52 % (24405)Termination phase: shuffling
% 2.70/1.52
% 2.70/1.52 % (24405)Memory used [KB]: 1407
% 2.70/1.52 % (24405)Time elapsed: 0.011 s
% 2.70/1.52 % (24405)Instructions burned: 18 (million)
% 2.70/1.52 % (24405)------------------------------
% 2.70/1.52 % (24405)------------------------------
% 2.70/1.52 % (24400)Instruction limit reached!
% 2.70/1.52 % (24400)------------------------------
% 2.70/1.52 % (24400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.52 % (24400)Termination reason: Unknown
% 2.70/1.52 % (24400)Termination phase: shuffling
% 2.70/1.52
% 2.70/1.52 % (24400)Memory used [KB]: 1279
% 2.70/1.52 % (24400)Time elapsed: 0.004 s
% 2.70/1.52 % (24400)Instructions burned: 5 (million)
% 2.70/1.52 % (24400)------------------------------
% 2.70/1.52 % (24400)------------------------------
% 2.70/1.52 % (24407)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_24206 for (2999ds/37Mi)
% 2.70/1.53 % (24409)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_24206 for (2999ds/3Mi)
% 2.70/1.53 % (24409)Instruction limit reached!
% 2.70/1.53 % (24409)------------------------------
% 2.70/1.53 % (24409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (24409)Termination reason: Unknown
% 2.70/1.53 % (24409)Termination phase: shuffling
% 2.70/1.53
% 2.70/1.53 % (24409)Memory used [KB]: 1151
% 2.70/1.53 % (24409)Time elapsed: 0.003 s
% 2.70/1.53 % (24409)Instructions burned: 3 (million)
% 2.70/1.53 % (24409)------------------------------
% 2.70/1.53 % (24409)------------------------------
% 2.70/1.53 % (24408)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_24206 for (2999ds/15Mi)
% 2.70/1.53 % (24407)Instruction limit reached!
% 2.70/1.53 % (24407)------------------------------
% 2.70/1.53 % (24407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (24407)Termination reason: Unknown
% 2.70/1.53 % (24407)Termination phase: Property scanning
% 2.70/1.53
% 2.70/1.53 % (24407)Memory used [KB]: 1663
% 2.70/1.53 % (24407)Time elapsed: 0.011 s
% 2.70/1.53 % (24407)Instructions burned: 39 (million)
% 2.70/1.53 % (24407)------------------------------
% 2.70/1.53 % (24407)------------------------------
% 2.70/1.54 % (24410)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_24206 for (2999ds/1041Mi)
% 2.70/1.54 % (24408)Instruction limit reached!
% 2.70/1.54 % (24408)------------------------------
% 2.70/1.54 % (24408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.54 % (24408)Termination reason: Unknown
% 2.70/1.54 % (24408)Termination phase: shuffling
% 2.70/1.54
% 2.70/1.54 % (24408)Memory used [KB]: 1407
% 2.70/1.54 % (24408)Time elapsed: 0.010 s
% 2.70/1.54 % (24408)Instructions burned: 16 (million)
% 2.70/1.54 % (24408)------------------------------
% 2.70/1.54 % (24408)------------------------------
% 2.70/1.54 % (24412)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/16Mi)
% 2.70/1.54 % (24413)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_24206 for (2999ds/3Mi)
% 2.70/1.55 % (24413)Instruction limit reached!
% 2.70/1.55 % (24413)------------------------------
% 2.70/1.55 % (24413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (24413)Termination reason: Unknown
% 2.70/1.55 % (24413)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (24413)Memory used [KB]: 1279
% 2.70/1.55 % (24413)Time elapsed: 0.002 s
% 2.70/1.55 % (24413)Instructions burned: 3 (million)
% 2.70/1.55 % (24413)------------------------------
% 2.70/1.55 % (24413)------------------------------
% 2.70/1.55 % (24411)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/7Mi)
% 2.70/1.55 % (24412)Instruction limit reached!
% 2.70/1.55 % (24412)------------------------------
% 2.70/1.55 % (24412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (24412)Termination reason: Unknown
% 2.70/1.55 % (24412)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (24412)Memory used [KB]: 1407
% 2.70/1.55 % (24412)Time elapsed: 0.008 s
% 2.70/1.55 % (24412)Instructions burned: 18 (million)
% 2.70/1.55 % (24412)------------------------------
% 2.70/1.55 % (24412)------------------------------
% 2.70/1.55 % (24411)Instruction limit reached!
% 2.70/1.55 % (24411)------------------------------
% 2.70/1.55 % (24411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (24411)Termination reason: Unknown
% 2.70/1.55 % (24411)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (24411)Memory used [KB]: 1279
% 2.70/1.55 % (24411)Time elapsed: 0.005 s
% 2.70/1.55 % (24411)Instructions burned: 7 (million)
% 2.70/1.55 % (24411)------------------------------
% 2.70/1.55 % (24411)------------------------------
% 2.70/1.55 % (24415)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/7Mi)
% 2.70/1.56 % (24415)Instruction limit reached!
% 2.70/1.56 % (24415)------------------------------
% 2.70/1.56 % (24415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (24415)Termination reason: Unknown
% 2.70/1.56 % (24415)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (24415)Memory used [KB]: 1279
% 2.70/1.56 % (24415)Time elapsed: 0.003 s
% 2.70/1.56 % (24415)Instructions burned: 7 (million)
% 2.70/1.56 % (24415)------------------------------
% 2.70/1.56 % (24415)------------------------------
% 2.70/1.56 % (24414)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_24206 for (2999ds/3Mi)
% 2.70/1.56 % (24416)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/3Mi)
% 2.70/1.56 % (24416)Instruction limit reached!
% 2.70/1.56 % (24416)------------------------------
% 2.70/1.56 % (24416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (24416)Termination reason: Unknown
% 2.70/1.56 % (24416)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (24416)Memory used [KB]: 1279
% 2.70/1.56 % (24416)Time elapsed: 0.003 s
% 2.70/1.56 % (24416)Instructions burned: 4 (million)
% 2.70/1.56 % (24416)------------------------------
% 2.70/1.56 % (24416)------------------------------
% 2.70/1.56 % (24414)Instruction limit reached!
% 2.70/1.56 % (24414)------------------------------
% 2.70/1.56 % (24414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (24414)Termination reason: Unknown
% 2.70/1.56 % (24414)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (24414)Memory used [KB]: 1279
% 2.70/1.56 % (24414)Time elapsed: 0.004 s
% 2.70/1.56 % (24414)Instructions burned: 4 (million)
% 2.70/1.56 % (24414)------------------------------
% 2.70/1.56 % (24414)------------------------------
% 2.70/1.57 % (24418)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/18Mi)
% 2.70/1.57 % (24417)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/4Mi)
% 2.70/1.57 % (24417)Instruction limit reached!
% 2.70/1.57 % (24417)------------------------------
% 2.70/1.57 % (24417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (24417)Termination reason: Unknown
% 2.70/1.57 % (24417)Termination phase: shuffling
% 2.70/1.57
% 2.70/1.57 % (24417)Memory used [KB]: 1279
% 2.70/1.57 % (24417)Time elapsed: 0.004 s
% 2.70/1.57 % (24417)Instructions burned: 4 (million)
% 2.70/1.57 % (24417)------------------------------
% 2.70/1.57 % (24417)------------------------------
% 2.70/1.57 % (24419)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/710Mi)
% 2.70/1.57 % (24418)Instruction limit reached!
% 2.70/1.57 % (24418)------------------------------
% 2.70/1.57 % (24418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (24418)Termination reason: Unknown
% 2.70/1.57 % (24418)Termination phase: shuffling
% 2.70/1.57
% 2.70/1.57 % (24418)Memory used [KB]: 1407
% 2.70/1.57 % (24418)Time elapsed: 0.007 s
% 2.70/1.57 % (24418)Instructions burned: 18 (million)
% 2.70/1.57 % (24418)------------------------------
% 2.70/1.57 % (24418)------------------------------
% 2.70/1.58 % (24420)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/6Mi)
% 2.70/1.58 % (24420)Instruction limit reached!
% 2.70/1.58 % (24420)------------------------------
% 2.70/1.58 % (24420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (24420)Termination reason: Unknown
% 2.70/1.58 % (24420)Termination phase: shuffling
% 2.70/1.58
% 2.70/1.58 % (24420)Memory used [KB]: 1279
% 2.70/1.58 % (24420)Time elapsed: 0.005 s
% 2.70/1.58 % (24420)Instructions burned: 6 (million)
% 2.70/1.58 % (24420)------------------------------
% 2.70/1.58 % (24420)------------------------------
% 2.70/1.58 % (24421)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_24206 for (2999ds/902Mi)
% 2.70/1.59 % (24399)Instruction limit reached!
% 2.70/1.59 % (24399)------------------------------
% 2.70/1.59 % (24399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (24399)Termination reason: Unknown
% 2.70/1.59 % (24399)Termination phase: Saturation
% 2.70/1.59
% 2.70/1.59 % (24399)Memory used [KB]: 6652
% 2.70/1.59 % (24399)Time elapsed: 0.069 s
% 2.70/1.59 % (24399)Instructions burned: 185 (million)
% 2.70/1.59 % (24399)------------------------------
% 2.70/1.59 % (24399)------------------------------
% 2.70/1.59 % (24422)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_24206 for (2999ds/21Mi)
% 2.70/1.59 % (24404)Instruction limit reached!
% 2.70/1.59 % (24404)------------------------------
% 2.70/1.59 % (24404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (24404)Termination reason: Unknown
% 2.70/1.59 % (24404)Termination phase: Saturation
% 2.70/1.59
% 2.70/1.59 % (24404)Memory used [KB]: 6652
% 2.70/1.59 % (24404)Time elapsed: 0.086 s
% 2.70/1.59 % (24404)Instructions burned: 278 (million)
% 2.70/1.59 % (24404)------------------------------
% 2.70/1.59 % (24404)------------------------------
% 2.70/1.60 % (24422)Instruction limit reached!
% 2.70/1.60 % (24422)------------------------------
% 2.70/1.60 % (24422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (24422)Termination reason: Unknown
% 2.70/1.60 % (24422)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (24422)Memory used [KB]: 1407
% 2.70/1.60 % (24422)Time elapsed: 0.009 s
% 2.70/1.60 % (24422)Instructions burned: 21 (million)
% 2.70/1.60 % (24422)------------------------------
% 2.70/1.60 % (24422)------------------------------
% 2.70/1.60 % (24423)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/5Mi)
% 2.70/1.60 % (24424)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_24206 for (2999ds/6Mi)
% 2.70/1.60 % (24423)Instruction limit reached!
% 2.70/1.60 % (24423)------------------------------
% 2.70/1.60 % (24423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (24423)Termination reason: Unknown
% 2.70/1.60 % (24423)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (24423)Memory used [KB]: 1279
% 2.70/1.60 % (24423)Time elapsed: 0.004 s
% 2.70/1.60 % (24423)Instructions burned: 5 (million)
% 2.70/1.60 % (24423)------------------------------
% 2.70/1.60 % (24423)------------------------------
% 2.70/1.60 % (24424)Instruction limit reached!
% 2.70/1.60 % (24424)------------------------------
% 2.70/1.60 % (24424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (24424)Termination reason: Unknown
% 2.70/1.60 % (24424)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (24424)Memory used [KB]: 1279
% 2.70/1.60 % (24424)Time elapsed: 0.003 s
% 2.70/1.60 % (24424)Instructions burned: 6 (million)
% 2.70/1.60 % (24424)------------------------------
% 2.70/1.60 % (24424)------------------------------
% 2.70/1.60 % (24425)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_24206 for (2999ds/377Mi)
% 2.70/1.61 % (24426)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_24206 for (2998ds/779Mi)
% 2.70/1.61 % (24427)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_24206 for (2998ds/19Mi)
% 2.70/1.62 % (24427)Instruction limit reached!
% 2.70/1.62 % (24427)------------------------------
% 2.70/1.62 % (24427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (24427)Termination reason: Unknown
% 2.70/1.62 % (24427)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (24427)Memory used [KB]: 1535
% 2.70/1.62 % (24427)Time elapsed: 0.008 s
% 2.70/1.62 % (24427)Instructions burned: 21 (million)
% 2.70/1.62 % (24427)------------------------------
% 2.70/1.62 % (24427)------------------------------
% 2.70/1.62 % (24428)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_24206 for (2998ds/879Mi)
% 2.70/1.63 % (24429)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_24206 for (2998ds/17Mi)
% 2.70/1.63 % (24429)Instruction limit reached!
% 2.70/1.63 % (24429)------------------------------
% 2.70/1.63 % (24429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (24429)Termination reason: Unknown
% 2.70/1.63 % (24429)Termination phase: shuffling
% 2.70/1.63
% 2.70/1.63 % (24429)Memory used [KB]: 1407
% 2.70/1.63 % (24429)Time elapsed: 0.007 s
% 2.70/1.63 % (24429)Instructions burned: 17 (million)
% 2.70/1.63 % (24429)------------------------------
% 2.70/1.63 % (24429)------------------------------
% 2.70/1.65 % (24410)First to succeed.
% 2.70/1.65 % (24430)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24206 for (2998ds/3Mi)
% 2.70/1.65 % (24430)Instruction limit reached!
% 2.70/1.65 % (24430)------------------------------
% 2.70/1.65 % (24430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (24430)Termination reason: Unknown
% 2.70/1.65 % (24430)Termination phase: shuffling
% 2.70/1.65
% 2.70/1.65 % (24430)Memory used [KB]: 1279
% 2.70/1.65 % (24430)Time elapsed: 0.004 s
% 2.70/1.65 % (24430)Instructions burned: 3 (million)
% 2.70/1.65 % (24430)------------------------------
% 2.70/1.65 % (24430)------------------------------
% 2.70/1.65 % (24410)Refutation found. Thanks to Tanya!
% 2.70/1.65 % SZS status Theorem for DTF2THF_24206
% 2.70/1.65 % SZS output start Proof for DTF2THF_24206
% 2.70/1.65 thf(type_def_5, type, partia1833973666xt_a_b: $tType).
% 2.70/1.65 thf(type_def_6, type, a: $tType).
% 2.70/1.65 thf(type_def_7, type, set_a: $tType).
% 2.70/1.65 thf(type_def_9, type, int: $tType).
% 2.70/1.65 thf(type_def_10, type, product_prod_a_a: $tType).
% 2.70/1.65 thf(func_def_0, type, partia1833973666xt_a_b: $tType).
% 2.70/1.65 thf(func_def_1, type, product_prod_a_a: $tType).
% 2.70/1.65 thf(func_def_2, type, set_a: $tType).
% 2.70/1.65 thf(func_def_3, type, int: $tType).
% 2.70/1.65 thf(func_def_4, type, a: $tType).
% 2.70/1.65 thf(func_def_5, type, a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a).
% 2.70/1.65 thf(func_def_6, type, additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o).
% 2.70/1.65 thf(func_def_7, type, set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a).
% 2.70/1.65 thf(func_def_8, type, partia1066395285xt_a_b: partia1833973666xt_a_b > set_a).
% 2.70/1.65 thf(func_def_9, type, mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a).
% 2.70/1.65 thf(func_def_10, type, minus_minus_int: int > int > int).
% 2.70/1.65 thf(func_def_11, type, minus_minus_set_a: set_a > set_a > set_a).
% 2.70/1.65 thf(func_def_12, type, plus_plus_int: int > int > int).
% 2.70/1.65 thf(func_def_13, type, uminus_uminus_int: int > int).
% 2.70/1.65 thf(func_def_14, type, uminus_uminus_set_a: set_a > set_a).
% 2.70/1.65 thf(func_def_15, type, locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o).
% 2.70/1.65 thf(func_def_16, type, locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o).
% 2.70/1.65 thf(func_def_17, type, locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o).
% 2.70/1.65 thf(func_def_18, type, locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o).
% 2.70/1.65 thf(func_def_19, type, bot_bot_set_a: set_a).
% 2.70/1.65 thf(func_def_20, type, ord_less_eq_int: int > int > $o).
% 2.70/1.65 thf(func_def_21, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.70/1.65 thf(func_def_22, type, product_fst_a_a: product_prod_a_a > a).
% 2.70/1.65 thf(func_def_23, type, product_snd_a_a: product_prod_a_a > a).
% 2.70/1.65 thf(func_def_24, type, a_inv_a_b: partia1833973666xt_a_b > a > a).
% 2.70/1.65 thf(func_def_25, type, a_minus_a_b: partia1833973666xt_a_b > a > a > a).
% 2.70/1.65 thf(func_def_26, type, add_pow_a_b_int: partia1833973666xt_a_b > int > a > a).
% 2.70/1.65 thf(func_def_27, type, add_a_b: partia1833973666xt_a_b > a > a > a).
% 2.70/1.65 thf(func_def_28, type, zero_a_b: partia1833973666xt_a_b > a).
% 2.70/1.65 thf(func_def_29, type, semiring_a_b: partia1833973666xt_a_b > $o).
% 2.70/1.65 thf(func_def_30, type, collect_a: (a > $o) > set_a).
% 2.70/1.65 thf(func_def_31, type, member_a: a > set_a > $o).
% 2.70/1.65 thf(func_def_32, type, r: partia1833973666xt_a_b).
% 2.70/1.65 thf(func_def_33, type, s: set_a).
% 2.70/1.65 thf(func_def_34, type, r2: a).
% 2.70/1.65 thf(func_def_35, type, r3: a).
% 2.70/1.65 thf(func_def_36, type, s2: a).
% 2.70/1.65 thf(func_def_37, type, s3: a).
% 2.70/1.65 thf(func_def_38, type, t: a).
% 2.70/1.65 thf(func_def_39, type, t2: a).
% 2.70/1.65 thf(func_def_40, type, x: product_prod_a_a).
% 2.70/1.65 thf(func_def_41, type, y: product_prod_a_a).
% 2.70/1.65 thf(func_def_58, type, ph1: !>[X0: $tType]:(X0)).
% 2.70/1.65 thf(f1246,plain,(
% 2.70/1.65 $false),
% 2.70/1.65 inference(trivial_inequality_removal,[status(thm)],[f1245])).
% 2.70/1.65 thf(f1245,plain,(
% 2.70/1.65 ($false = $true)),
% 2.70/1.65 inference(boolean_simplification,[status(thm)],[f1244])).
% 2.70/1.65 thf(f1244,plain,(
% 2.70/1.65 ((~ $true) = $true)),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1115,f1243])).
% 2.70/1.65 thf(f1243,plain,(
% 2.70/1.65 (((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))) = $true)),
% 2.70/1.65 inference(boolean_simplification,[status(thm)],[f1242])).
% 2.70/1.65 thf(f1242,plain,(
% 2.70/1.65 (($true => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1134,f1233])).
% 2.70/1.65 thf(f1233,plain,(
% 2.70/1.65 ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(boolean_simplification,[status(thm)],[f1224])).
% 2.70/1.65 thf(f1224,plain,(
% 2.70/1.65 (($true => (member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r))) = $true)),
% 2.70/1.65 inference(superposition,[status(thm)],[f1206,f781])).
% 2.70/1.65 thf(f781,plain,(
% 2.70/1.65 ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ s) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f356])).
% 2.70/1.65 thf(f356,plain,(
% 2.70/1.65 ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ s) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f355])).
% 2.70/1.65 thf(f355,plain,(
% 2.70/1.65 (member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ s)),
% 2.70/1.65 inference(rectify,[status(thm)],[f66])).
% 2.70/1.65 thf(f66,axiom,(
% 2.70/1.65 (member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ s)),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_65_f8)).
% 2.70/1.65 thf(f1206,plain,(
% 2.70/1.65 ( ! [X1 : a] : ((((member_a @ X1 @ s) => (member_a @ X1 @ (partia1066395285xt_a_b @ r))) = $true)) )),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f1205])).
% 2.70/1.65 thf(f1205,plain,(
% 2.70/1.65 ( ! [X1 : a] : (($true = ((^[Y0 : a]: ((member_a @ Y0 @ s) => (member_a @ Y0 @ (partia1066395285xt_a_b @ r)))) @ X1))) )),
% 2.70/1.65 inference(pi_clausification,[status(thm)],[f1072])).
% 2.70/1.65 thf(f1072,plain,(
% 2.70/1.65 ((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ s) => (member_a @ Y0 @ (partia1066395285xt_a_b @ r))))) = $true)),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f1008])).
% 2.70/1.65 thf(f1008,plain,(
% 2.70/1.65 ($true = ((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ s @ (partia1066395285xt_a_b @ r)))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f884,f857])).
% 2.70/1.65 thf(f857,plain,(
% 2.70/1.65 (ord_less_eq_set_a = (^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f631])).
% 2.70/1.65 thf(f631,plain,(
% 2.70/1.65 (ord_less_eq_set_a = (^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f630])).
% 2.70/1.65 thf(f630,plain,(
% 2.70/1.65 ((^[X0 : set_a, X1 : set_a] : (! [X2 : a] : ((member_a @ X2 @ X0) => (member_a @ X2 @ X1)))) = ord_less_eq_set_a)),
% 2.70/1.65 inference(rectify,[status(thm)],[f244])).
% 2.70/1.65 thf(f244,axiom,(
% 2.70/1.65 ((^[X44 : set_a, X45 : set_a] : (! [X9 : a] : ((member_a @ X9 @ X44) => (member_a @ X9 @ X45)))) = ord_less_eq_set_a)),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_243_subset__eq)).
% 2.70/1.65 thf(f884,plain,(
% 2.70/1.65 ((ord_less_eq_set_a @ s @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f648])).
% 2.70/1.65 thf(f648,plain,(
% 2.70/1.65 ((ord_less_eq_set_a @ s @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f647])).
% 2.70/1.65 thf(f647,plain,(
% 2.70/1.65 (ord_less_eq_set_a @ s @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 inference(rectify,[status(thm)],[f143])).
% 2.70/1.65 thf(f143,axiom,(
% 2.70/1.65 (ord_less_eq_set_a @ s @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_142_subset)).
% 2.70/1.65 thf(f1134,plain,(
% 2.70/1.65 (((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1128,f1133])).
% 2.70/1.65 thf(f1133,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1132,f1074])).
% 2.70/1.65 thf(f1074,plain,(
% 2.70/1.65 ((zero_a_b @ r) = (mult_a_ring_ext_a_b @ r @ t @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f1022])).
% 2.70/1.65 thf(f1022,plain,(
% 2.70/1.65 ((zero_a_b @ r) = (mult_a_ring_ext_a_b @ r @ t @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f934,f890])).
% 2.70/1.65 thf(f890,plain,(
% 2.70/1.65 (a_minus_a_b = (^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f577])).
% 2.70/1.65 thf(f577,plain,(
% 2.70/1.65 (a_minus_a_b = (^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f576])).
% 2.70/1.65 thf(f576,plain,(
% 2.70/1.65 ((^[X0 : partia1833973666xt_a_b, X1 : a, X2 : a] : (add_a_b @ X0 @ X1 @ (a_inv_a_b @ X0 @ X2))) = a_minus_a_b)),
% 2.70/1.65 inference(rectify,[status(thm)],[f131])).
% 2.70/1.65 thf(f131,axiom,(
% 2.70/1.65 ((^[X30 : partia1833973666xt_a_b, X9 : a, X31 : a] : (add_a_b @ X30 @ X9 @ (a_inv_a_b @ X30 @ X31))) = a_minus_a_b)),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_130_a__minus__def)).
% 2.70/1.65 thf(f934,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))) = (zero_a_b @ r))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f361])).
% 2.70/1.65 thf(f361,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))) = (zero_a_b @ r))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f59])).
% 2.70/1.65 thf(f59,axiom,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))) = (zero_a_b @ r))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_58_f7)).
% 2.70/1.65 thf(f1132,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y))))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1131,f1073])).
% 2.70/1.65 thf(f1073,plain,(
% 2.70/1.65 ((zero_a_b @ r) = (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f1014])).
% 2.70/1.65 thf(f1014,plain,(
% 2.70/1.65 ((zero_a_b @ r) = (mult_a_ring_ext_a_b @ r @ t2 @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f910,f890])).
% 2.70/1.65 thf(f910,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))) = (zero_a_b @ r))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f305])).
% 2.70/1.65 thf(f305,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))) = (zero_a_b @ r))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f58])).
% 2.70/1.65 thf(f58,axiom,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))) = (zero_a_b @ r))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_57_f5)).
% 2.70/1.65 thf(f1131,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y))))))))),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1042,f1120])).
% 2.70/1.65 thf(f1120,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))))),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1071,f1119])).
% 2.70/1.65 thf(f1119,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))))),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1107,f1073])).
% 2.70/1.65 thf(f1107,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f990])).
% 2.70/1.65 thf(f990,plain,(
% 2.70/1.65 (((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f848,f890,f890])).
% 2.70/1.65 thf(f848,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f312])).
% 2.70/1.65 thf(f312,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f42])).
% 2.70/1.65 thf(f42,axiom,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_41_f25)).
% 2.70/1.65 thf(f1071,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f972])).
% 2.70/1.65 thf(f972,plain,(
% 2.70/1.65 (((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f776,f890,f890])).
% 2.70/1.65 thf(f776,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f290])).
% 2.70/1.65 thf(f290,plain,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f43])).
% 2.70/1.65 thf(f43,axiom,(
% 2.70/1.65 ((mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) = (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_42_f32)).
% 2.70/1.65 thf(f1042,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y))))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f971])).
% 2.70/1.65 thf(f971,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f768,f890,f890,f890,f890])).
% 2.70/1.65 thf(f768,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f344])).
% 2.70/1.65 thf(f344,plain,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f20])).
% 2.70/1.65 thf(f20,axiom,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_19_f40)).
% 2.70/1.65 thf(f1128,plain,(
% 2.70/1.65 (((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1127,f1120])).
% 2.70/1.65 thf(f1127,plain,(
% 2.70/1.65 (((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(boolean_simplification,[status(thm)],[f1126])).
% 2.70/1.65 thf(f1126,plain,(
% 2.70/1.65 (($true => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))) = $true)),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1125,f1081])).
% 2.70/1.65 thf(f1081,plain,(
% 2.70/1.65 ($true = (member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))) @ (partia1066395285xt_a_b @ r)))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f980])).
% 2.70/1.65 thf(f980,plain,(
% 2.70/1.65 ((member_a @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f805,f890])).
% 2.70/1.65 thf(f805,plain,(
% 2.70/1.65 ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f255])).
% 2.70/1.65 thf(f255,plain,(
% 2.70/1.65 ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f254])).
% 2.70/1.65 thf(f254,plain,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 inference(rectify,[status(thm)],[f14])).
% 2.70/1.65 thf(f14,axiom,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_13_f42)).
% 2.70/1.65 thf(f1125,plain,(
% 2.70/1.65 (((member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))) = $true)),
% 2.70/1.65 inference(boolean_simplification,[status(thm)],[f1124])).
% 2.70/1.65 thf(f1124,plain,(
% 2.70/1.65 (($true => ((member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))))) = $true)),
% 2.70/1.65 inference(forward_demodulation,[status(thm)],[f1056,f1062])).
% 2.70/1.65 thf(f1062,plain,(
% 2.70/1.65 ((member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f946])).
% 2.70/1.65 thf(f946,plain,(
% 2.70/1.65 ($true = (member_a @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)))),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f701,f890])).
% 2.70/1.65 thf(f701,plain,(
% 2.70/1.65 ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f261])).
% 2.70/1.65 thf(f261,plain,(
% 2.70/1.65 ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f260])).
% 2.70/1.65 thf(f260,plain,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 inference(rectify,[status(thm)],[f13])).
% 2.70/1.65 thf(f13,axiom,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_12_f41)).
% 2.70/1.65 thf(f1056,plain,(
% 2.70/1.65 (((member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))))) = $true)),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f995])).
% 2.70/1.65 thf(f995,plain,(
% 2.70/1.65 (((member_a @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))) = $true)),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f860,f890,f890,f890,f890,f890,f890])).
% 2.70/1.65 thf(f860,plain,(
% 2.70/1.65 (((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f570])).
% 2.70/1.65 thf(f570,plain,(
% 2.70/1.65 (((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f569])).
% 2.70/1.65 thf(f569,plain,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))),
% 2.70/1.65 inference(rectify,[status(thm)],[f22])).
% 2.70/1.65 thf(f22,axiom,(
% 2.70/1.65 (member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))) @ (partia1066395285xt_a_b @ r)) => ((member_a @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (partia1066395285xt_a_b @ r)) => ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062)).
% 2.70/1.65 thf(f1115,plain,(
% 2.70/1.65 ((~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (zero_a_b @ r)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1114,f1073])).
% 2.70/1.65 thf(f1114,plain,(
% 2.70/1.65 ((~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (zero_a_b @ r))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))) = $true)),
% 2.70/1.65 inference(backward_demodulation,[status(thm)],[f1076,f1074])).
% 2.70/1.65 thf(f1076,plain,(
% 2.70/1.65 ($true = (~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x)))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y))))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x))))) @ (add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (a_inv_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y))))))))))),
% 2.70/1.65 inference(beta_eta_normalization,[status(thm)],[f1020])).
% 2.70/1.65 thf(f1020,plain,(
% 2.70/1.65 ((~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ ((^[Y0 : partia1833973666xt_a_b]: ((^[Y1 : a]: ((^[Y2 : a]: (add_a_b @ Y0 @ Y1 @ (a_inv_a_b @ Y0 @ Y2))))))) @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))) = $true)),
% 2.70/1.65 inference(definition_unfolding,[status(thm)],[f928,f890,f890,f890,f890])).
% 2.70/1.65 thf(f928,plain,(
% 2.70/1.65 ((~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))) = $true)),
% 2.70/1.65 inference(cnf_transformation,[status(thm)],[f420])).
% 2.70/1.65 thf(f420,plain,(
% 2.70/1.65 ((~ ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))) = $true)),
% 2.70/1.65 inference(fool_elimination,[status(thm)],[f246])).
% 2.70/1.65 thf(f246,negated_conjecture,(
% 2.70/1.65 ~((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 inference(negated_conjecture,[status(cth)],[f245])).
% 2.70/1.65 thf(f245,conjecture,(
% 2.70/1.65 ((add_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t @ s3) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ t2 @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ r2) @ (mult_a_ring_ext_a_b @ r @ s2 @ (product_fst_a_a @ x))))) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ s2) @ (product_snd_a_a @ x)) @ (mult_a_ring_ext_a_b @ r @ t @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ (product_fst_a_a @ y)))))) = (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ t2 @ t) @ (add_a_b @ r @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s3 @ r2)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ y) @ (product_fst_a_a @ x)))) @ (a_minus_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_snd_a_a @ y)) @ (mult_a_ring_ext_a_b @ r @ s2 @ r3)) @ (mult_a_ring_ext_a_b @ r @ (mult_a_ring_ext_a_b @ r @ s2 @ s3) @ (mult_a_ring_ext_a_b @ r @ (product_snd_a_a @ x) @ (product_fst_a_a @ y)))))))),
% 2.70/1.65 file('/export/starexec/sandbox/tmp/tmp.uc6feKMVkK/DTF2THF_24206.p',conj_0)).
% 2.70/1.65 % SZS output end Proof for DTF2THF_24206
% 2.70/1.65 % (24410)------------------------------
% 2.70/1.65 % (24410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (24410)Termination reason: Refutation
% 2.70/1.65
% 2.70/1.65 % (24410)Memory used [KB]: 7547
% 2.70/1.65 % (24410)Time elapsed: 0.115 s
% 2.70/1.65 % (24410)Instructions burned: 261 (million)
% 2.70/1.65 % (24410)------------------------------
% 2.70/1.65 % (24410)------------------------------
% 2.70/1.65 % (24398)Success in time 0.156 s
%------------------------------------------------------------------------------