TPTP Problem File: SLH0767^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0014_Set_Multiset_Extras/prob_00264_010259__27894344_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1383 ( 578 unt; 109 typ; 0 def)
% Number of atoms : 4221 (1775 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 16762 ( 442 ~; 63 |; 442 &;13728 @)
% ( 0 <=>;2087 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 1052 (1052 >; 0 *; 0 +; 0 <<)
% Number of symbols : 102 ( 99 usr; 13 con; 0-4 aty)
% Number of variables : 4226 ( 470 ^;3669 !; 87 ?;4226 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:45:25.482
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (99)
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
finite_finite_num: set_num > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
finite_finite_set_b: set_set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Omonoid_001tf__a,type,
monoid_a: ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001tf__a_001t__Nat__Onat,type,
groups5773243554134465322_a_nat: ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001tf__a_001tf__a,type,
groups1779759026887736868um_a_a: ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001tf__a_001tf__b,type,
groups1779759026887736869um_a_b: ( a > a > a ) > a > ( b > a ) > set_b > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_H_001tf__a_001t__Nat__Onat,type,
groups6420517015690499521_a_nat: ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_H_001tf__a_001tf__a,type,
groups8906383913375152973um_a_a: ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_H_001tf__a_001tf__b,type,
groups8906383913375152974um_a_b: ( a > a > a ) > a > ( b > a ) > set_b > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_001tf__a_001t__Nat__Onat,type,
groups1957776620359388557_a_nat: ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_001tf__a_001tf__a,type,
groups2061451144089001601od_a_a: ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_001tf__a_001tf__b,type,
groups2061451144089001602od_a_b: ( a > a > a ) > a > ( b > a ) > set_b > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_H_001tf__a_001t__Nat__Onat,type,
groups7824906719281202852_a_nat: ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_H_001tf__a_001tf__a,type,
groups4667919067926330666od_a_a: ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult_Oprod_H_001tf__a_001tf__b,type,
groups4667919067926330667od_a_b: ( a > a > a ) > a > ( b > a ) > set_b > a ).
thf(sy_c_Groups__List_Ocomm__monoid__list_001tf__a,type,
groups1759187151362946711list_a: ( a > a > a ) > a > $o ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set_001tf__a,type,
groups8881925628872693537_set_a: ( a > a > a ) > a > $o ).
thf(sy_c_Groups__List_Omonoid__list_001tf__a,type,
groups_monoid_list_a: ( a > a > a ) > a > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Multiset_Ocomm__monoid__mset_001tf__a,type,
comm_monoid_mset_a: ( a > a > a ) > a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1_ONats_001tf__a,type,
semiring_Nats_a: a > ( a > a > a ) > a > set_a ).
thf(sy_c_Nat_Osemiring__1_Oof__nat__aux_001tf__a,type,
semiri4123672225490436309_aux_a: ( a > a ) > nat > a > a ).
thf(sy_c_Num_Oneg__numeral_Odbl_001tf__a,type,
neg_dbl_a: ( a > a > a ) > a > a ).
thf(sy_c_Num_Oneg__numeral_Odbl__inc_001tf__a,type,
neg_dbl_inc_a: ( a > a > a ) > a > a > a ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Oring__1_Oiszero_001tf__a,type,
ring_iszero_a: a > a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
bot_bot_set_num: set_num ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Power_Opower_Opower_001tf__a,type,
power_a: a > ( a > a > a ) > a > nat > a ).
thf(sy_c_Rings_Ozero__neq__one_Oof__bool_001tf__a,type,
zero_neq_of_bool_a: a > a > $o > a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_A,type,
a2: set_b ).
thf(sy_v_f,type,
f: b > a ).
thf(sy_v_one,type,
one2: a ).
thf(sy_v_plus,type,
plus: a > a > a ).
thf(sy_v_times,type,
times: a > a > a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1268)
thf(fact_0_local_Oadd_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus @ B @ ( plus @ A @ C ) )
= ( plus @ A @ ( plus @ B @ C ) ) ) ).
% local.add.left_commute
thf(fact_1_add__commute,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% add_commute
thf(fact_2_local_Oadd__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus @ A @ B )
= ( plus @ A @ C ) )
=> ( B = C ) ) ).
% local.add_left_imp_eq
thf(fact_3_local_Oadd__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus @ B @ A )
= ( plus @ C @ A ) )
=> ( B = C ) ) ).
% local.add_right_imp_eq
thf(fact_4_local_Omult_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( times @ B @ ( times @ A @ C ) )
= ( times @ A @ ( times @ B @ C ) ) ) ).
% local.mult.left_commute
thf(fact_5_mult__assoc,axiom,
! [A: a,B: a,C: a] :
( ( times @ ( times @ A @ B ) @ C )
= ( times @ A @ ( times @ B @ C ) ) ) ).
% mult_assoc
thf(fact_6_local_Ozero__neq__one,axiom,
zero != one2 ).
% local.zero_neq_one
thf(fact_7_local_Ocombine__common__factor,axiom,
! [A: a,E: a,B: a,C: a] :
( ( plus @ ( times @ A @ E ) @ ( plus @ ( times @ B @ E ) @ C ) )
= ( plus @ ( times @ ( plus @ A @ B ) @ E ) @ C ) ) ).
% local.combine_common_factor
thf(fact_8_local_Odistrib__left,axiom,
! [A: a,B: a,C: a] :
( ( times @ A @ ( plus @ B @ C ) )
= ( plus @ ( times @ A @ B ) @ ( times @ A @ C ) ) ) ).
% local.distrib_left
thf(fact_9_local_Omult__not__zero,axiom,
! [A: a,B: a] :
( ( ( times @ A @ B )
!= zero )
=> ( ( A != zero )
& ( B != zero ) ) ) ).
% local.mult_not_zero
thf(fact_10_local_Osum_Ocong,axiom,
! [A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( A2 = B2 )
=> ( ! [X: a] :
( ( member_a @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ B2 ) ) ) ) ).
% local.sum.cong
thf(fact_11_local_Osum_Ocong,axiom,
! [A2: set_b,B2: set_b,G: b > a,H: b > a] :
( ( A2 = B2 )
=> ( ! [X: b] :
( ( member_b @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ B2 ) ) ) ) ).
% local.sum.cong
thf(fact_12_local_Osum_Ocong,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ B2 ) ) ) ) ).
% local.sum.cong
thf(fact_13_local_Osum_Oeq__general,axiom,
! [B2: set_a,A2: set_a,H: a > a,Gamma: a > a,Phi: a > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_14_local_Osum_Oeq__general,axiom,
! [B2: set_b,A2: set_a,H: a > b,Gamma: b > a,Phi: a > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_15_local_Osum_Oeq__general,axiom,
! [B2: set_nat,A2: set_a,H: a > nat,Gamma: nat > a,Phi: a > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_16_local_Osum_Oeq__general,axiom,
! [B2: set_a,A2: set_b,H: b > a,Gamma: a > a,Phi: b > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_17_local_Osum_Oeq__general,axiom,
! [B2: set_b,A2: set_b,H: b > b,Gamma: b > a,Phi: b > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_18_local_Osum_Oeq__general,axiom,
! [B2: set_nat,A2: set_b,H: b > nat,Gamma: nat > a,Phi: b > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_19_local_Osum_Oeq__general,axiom,
! [B2: set_a,A2: set_nat,H: nat > a,Gamma: a > a,Phi: nat > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_20_local_Osum_Oeq__general,axiom,
! [B2: set_b,A2: set_nat,H: nat > b,Gamma: b > a,Phi: nat > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_21_local_Osum_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general
thf(fact_22_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > a,A2: set_a,H: a > a,Gamma: a > a,Phi: a > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ( ( member_a @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_23_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_b,K: b > a,A2: set_a,H: a > b,Gamma: b > a,Phi: a > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ( ( member_a @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_24_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > a,A2: set_a,H: a > nat,Gamma: nat > a,Phi: a > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( ( member_a @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_25_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > b,A2: set_b,H: b > a,Gamma: a > a,Phi: b > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ( ( member_b @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_26_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_b,K: b > b,A2: set_b,H: b > b,Gamma: b > a,Phi: b > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ( ( member_b @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_27_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > b,A2: set_b,H: b > nat,Gamma: nat > a,Phi: b > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( ( member_b @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_28_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > nat,A2: set_nat,H: nat > a,Gamma: a > a,Phi: nat > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ( ( member_nat @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_29_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_b,K: b > nat,A2: set_nat,H: nat > b,Gamma: b > a,Phi: nat > a] :
( ! [Y: b] :
( ( member_b @ Y @ B2 )
=> ( ( member_nat @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_b @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_30_local_Osum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( ( member_nat @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ Phi @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ Gamma @ B2 ) ) ) ) ).
% local.sum.eq_general_inverses
thf(fact_31_local_Osum_Oneutral,axiom,
! [A2: set_b,G: b > a] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( G @ X )
= zero ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= zero ) ) ).
% local.sum.neutral
thf(fact_32_local_Osum_Oneutral,axiom,
! [A2: set_nat,G: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( G @ X )
= zero ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= zero ) ) ).
% local.sum.neutral
thf(fact_33_local_Osum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > a,A2: set_a] :
( ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 )
!= zero )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( G @ A3 )
= zero ) ) ) ).
% local.sum.not_neutral_contains_not_neutral
thf(fact_34_local_Osum_Onot__neutral__contains__not__neutral,axiom,
! [G: b > a,A2: set_b] :
( ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
!= zero )
=> ~ ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( G @ A3 )
= zero ) ) ) ).
% local.sum.not_neutral_contains_not_neutral
thf(fact_35_local_Osum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > a,A2: set_nat] :
( ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
!= zero )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero ) ) ) ).
% local.sum.not_neutral_contains_not_neutral
thf(fact_36_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_a,I: a > a,J: a > a,T: set_a,H: a > a,G: a > a] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_37_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_a,I: b > a,J: a > b,T: set_b,H: b > a,G: a > a] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_b @ ( J @ A3 ) @ T ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_38_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_a,I: nat > a,J: a > nat,T: set_nat,H: nat > a,G: a > a] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_39_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_b,I: a > b,J: b > a,T: set_a,H: a > a,G: b > a] :
( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_b @ ( I @ B3 ) @ S ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_40_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_b,I: b > b,J: b > b,T: set_b,H: b > a,G: b > a] :
( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( member_b @ ( J @ A3 ) @ T ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( member_b @ ( I @ B3 ) @ S ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_41_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_b,I: nat > b,J: b > nat,T: set_nat,H: nat > a,G: b > a] :
( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_b @ ( I @ B3 ) @ S ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_42_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: a > nat,J: nat > a,T: set_a,H: a > a,G: nat > a] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_43_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: b > nat,J: nat > b,T: set_b,H: b > a,G: nat > a] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_b @ ( J @ A3 ) @ T ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_44_local_Osum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H: nat > a,G: nat > a] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness
thf(fact_45_local_Oleft__right__inverse__power,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( times @ X3 @ Y2 )
= one2 )
=> ( ( times @ ( power_a @ one2 @ times @ X3 @ N ) @ ( power_a @ one2 @ times @ Y2 @ N ) )
= one2 ) ) ).
% local.left_right_inverse_power
thf(fact_46_local_Opower__commutes,axiom,
! [A: a,N: nat] :
( ( times @ ( power_a @ one2 @ times @ A @ N ) @ A )
= ( times @ A @ ( power_a @ one2 @ times @ A @ N ) ) ) ).
% local.power_commutes
thf(fact_47_local_Opower__commuting__commutes,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( times @ X3 @ Y2 )
= ( times @ Y2 @ X3 ) )
=> ( ( times @ ( power_a @ one2 @ times @ X3 @ N ) @ Y2 )
= ( times @ Y2 @ ( power_a @ one2 @ times @ X3 @ N ) ) ) ) ).
% local.power_commuting_commutes
thf(fact_48_local_Opower__mult__distrib,axiom,
! [A: a,B: a,N: nat] :
( ( power_a @ one2 @ times @ ( times @ A @ B ) @ N )
= ( times @ ( power_a @ one2 @ times @ A @ N ) @ ( power_a @ one2 @ times @ B @ N ) ) ) ).
% local.power_mult_distrib
thf(fact_49_local_Osum_Odistrib,axiom,
! [G: b > a,H: b > a,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] : ( plus @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ H @ A2 ) ) ) ).
% local.sum.distrib
thf(fact_50_local_Osum_Odistrib,axiom,
! [G: nat > a,H: nat > a,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] : ( plus @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ H @ A2 ) ) ) ).
% local.sum.distrib
thf(fact_51_local_Osum_Oswap,axiom,
! [G: b > b > a,B2: set_b,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% local.sum.swap
thf(fact_52_local_Osum_Oswap,axiom,
! [G: b > nat > a,B2: set_nat,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% local.sum.swap
thf(fact_53_local_Osum_Oswap,axiom,
! [G: nat > b > a,B2: set_b,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% local.sum.swap
thf(fact_54_local_Osum_Oswap,axiom,
! [G: nat > nat > a,B2: set_nat,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% local.sum.swap
thf(fact_55_local_Odouble__sum__split__case2,axiom,
! [G: a > a > a,A2: set_a] :
( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [I2: a] : ( groups1779759026887736868um_a_a @ plus @ zero @ ( G @ I2 ) @ A2 )
@ A2 )
= ( plus
@ ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [I2: a] : ( G @ I2 @ I2 )
@ A2 )
@ ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [I2: a] :
( groups1779759026887736868um_a_a @ plus @ zero @ ( G @ I2 )
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( A4 != I2 ) ) ) )
@ A2 ) ) ) ).
% local.double_sum_split_case2
thf(fact_56_local_Odouble__sum__split__case2,axiom,
! [G: b > b > a,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ I2 ) @ A2 )
@ A2 )
= ( plus
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( G @ I2 @ I2 )
@ A2 )
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ I2 )
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( A4 != I2 ) ) ) )
@ A2 ) ) ) ).
% local.double_sum_split_case2
thf(fact_57_local_Odouble__sum__split__case2,axiom,
! [G: nat > nat > a,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ I2 ) @ A2 )
@ A2 )
= ( plus
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ I2 @ I2 )
@ A2 )
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ I2 )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( A4 != I2 ) ) ) )
@ A2 ) ) ) ).
% local.double_sum_split_case2
thf(fact_58_local_Osum__reorder__triple,axiom,
! [G: b > b > b > a,C2: set_b,B2: set_b,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_59_local_Osum__reorder__triple,axiom,
! [G: b > b > nat > a,C2: set_nat,B2: set_b,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_60_local_Osum__reorder__triple,axiom,
! [G: b > nat > b > a,C2: set_b,B2: set_nat,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_61_local_Osum__reorder__triple,axiom,
! [G: b > nat > nat > a,C2: set_nat,B2: set_nat,A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [L: b] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_62_local_Osum__reorder__triple,axiom,
! [G: nat > b > b > a,C2: set_b,B2: set_b,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_63_local_Osum__reorder__triple,axiom,
! [G: nat > b > nat > a,C2: set_nat,B2: set_b,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_64_local_Osum__reorder__triple,axiom,
! [G: nat > nat > b > a,C2: set_b,B2: set_nat,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_65_local_Osum__reorder__triple,axiom,
! [G: nat > nat > nat > a,C2: set_nat,B2: set_nat,A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ L @ I2 ) @ C2 )
@ B2 )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [L: nat] : ( G @ L @ I2 @ J2 )
@ A2 )
@ C2 )
@ B2 ) ) ).
% local.sum_reorder_triple
thf(fact_66_local_Osum__distrib__left,axiom,
! [R: a,F: b > a,A2: set_b] :
( ( times @ R @ ( groups1779759026887736869um_a_b @ plus @ zero @ F @ A2 ) )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [N2: b] : ( times @ R @ ( F @ N2 ) )
@ A2 ) ) ).
% local.sum_distrib_left
thf(fact_67_local_Osum__distrib__left,axiom,
! [R: a,F: nat > a,A2: set_nat] :
( ( times @ R @ ( groups5773243554134465322_a_nat @ plus @ zero @ F @ A2 ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [N2: nat] : ( times @ R @ ( F @ N2 ) )
@ A2 ) ) ).
% local.sum_distrib_left
thf(fact_68_local_Osum__distrib__right,axiom,
! [F: b > a,A2: set_b,R: a] :
( ( times @ ( groups1779759026887736869um_a_b @ plus @ zero @ F @ A2 ) @ R )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [N2: b] : ( times @ ( F @ N2 ) @ R )
@ A2 ) ) ).
% local.sum_distrib_right
thf(fact_69_local_Osum__distrib__right,axiom,
! [F: nat > a,A2: set_nat,R: a] :
( ( times @ ( groups5773243554134465322_a_nat @ plus @ zero @ F @ A2 ) @ R )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [N2: nat] : ( times @ ( F @ N2 ) @ R )
@ A2 ) ) ).
% local.sum_distrib_right
thf(fact_70_local_Opower__numeral__even,axiom,
! [Z: a,W: num] :
( ( power_a @ one2 @ times @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times @ ( power_a @ one2 @ times @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_a @ one2 @ times @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% local.power_numeral_even
thf(fact_71_local_Oadd__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus @ A @ B )
= ( plus @ A @ C ) )
= ( B = C ) ) ).
% local.add_left_cancel
thf(fact_72_local_Oadd__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus @ B @ A )
= ( plus @ C @ A ) )
= ( B = C ) ) ).
% local.add_right_cancel
thf(fact_73_local_Oone__power2,axiom,
( ( power_a @ one2 @ times @ one2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one2 ) ).
% local.one_power2
thf(fact_74_local_Opower2__eq__square,axiom,
! [A: a] :
( ( power_a @ one2 @ times @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times @ A @ A ) ) ).
% local.power2_eq_square
thf(fact_75_local_Opower4__eq__xxxx,axiom,
! [X3: a] :
( ( power_a @ one2 @ times @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times @ ( times @ ( times @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% local.power4_eq_xxxx
thf(fact_76_local_Ozero__power2,axiom,
( ( power_a @ one2 @ times @ zero @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero ) ).
% local.zero_power2
thf(fact_77_local_Oadd__cancel__left__left,axiom,
! [B: a,A: a] :
( ( ( plus @ B @ A )
= A )
= ( B = zero ) ) ).
% local.add_cancel_left_left
thf(fact_78_local_Oadd__cancel__left__right,axiom,
! [A: a,B: a] :
( ( ( plus @ A @ B )
= A )
= ( B = zero ) ) ).
% local.add_cancel_left_right
thf(fact_79_local_Oadd__cancel__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus @ B @ A ) )
= ( B = zero ) ) ).
% local.add_cancel_right_left
thf(fact_80_local_Oadd__cancel__right__right,axiom,
! [A: a,B: a] :
( ( A
= ( plus @ A @ B ) )
= ( B = zero ) ) ).
% local.add_cancel_right_right
thf(fact_81_local_Opower__one,axiom,
! [N: nat] :
( ( power_a @ one2 @ times @ one2 @ N )
= one2 ) ).
% local.power_one
thf(fact_82_assms,axiom,
finite_finite_b @ a2 ).
% assms
thf(fact_83_local_Osum_Oneutral__const,axiom,
! [A2: set_b] :
( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [Uu: b] : zero
@ A2 )
= zero ) ).
% local.sum.neutral_const
thf(fact_84_local_Osum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [Uu: nat] : zero
@ A2 )
= zero ) ).
% local.sum.neutral_const
thf(fact_85_local_Osum_Oempty,axiom,
! [G: b > a] :
( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ bot_bot_set_b )
= zero ) ).
% local.sum.empty
thf(fact_86_local_Osum_Oempty,axiom,
! [G: nat > a] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ bot_bot_set_nat )
= zero ) ).
% local.sum.empty
thf(fact_87_local_Opower__zero__numeral,axiom,
! [K: num] :
( ( power_a @ one2 @ times @ zero @ ( numeral_numeral_nat @ K ) )
= zero ) ).
% local.power_zero_numeral
thf(fact_88_local_Odbl__inc__def,axiom,
! [X3: a] :
( ( neg_dbl_inc_a @ plus @ one2 @ X3 )
= ( plus @ ( plus @ X3 @ X3 ) @ one2 ) ) ).
% local.dbl_inc_def
thf(fact_89_local_Oprod_Ocomm__monoid__list__set__axioms,axiom,
groups8881925628872693537_set_a @ times @ one2 ).
% local.prod.comm_monoid_list_set_axioms
thf(fact_90_local_Osum_Ocomm__monoid__list__set__axioms,axiom,
groups8881925628872693537_set_a @ plus @ zero ).
% local.sum.comm_monoid_list_set_axioms
thf(fact_91_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_95_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_96_insert__filter__set__true,axiom,
! [P: b > $o,X3: b,A2: set_b] :
( ( P @ X3 )
=> ( ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ ( insert_b @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( insert_b @ X3
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ) ).
% insert_filter_set_true
thf(fact_97_insert__filter__set__true,axiom,
! [P: a > $o,X3: a,A2: set_a] :
( ( P @ X3 )
=> ( ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ ( insert_a @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( insert_a @ X3
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ) ).
% insert_filter_set_true
thf(fact_98_insert__filter__set__true,axiom,
! [P: nat > $o,X3: nat,A2: set_nat] :
( ( P @ X3 )
=> ( ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ ( insert_nat @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( insert_nat @ X3
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ) ).
% insert_filter_set_true
thf(fact_99_insert__filter__set__false,axiom,
! [P: b > $o,X3: b,A2: set_b] :
( ~ ( P @ X3 )
=> ( ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ ( insert_b @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ).
% insert_filter_set_false
thf(fact_100_insert__filter__set__false,axiom,
! [P: a > $o,X3: a,A2: set_a] :
( ~ ( P @ X3 )
=> ( ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ ( insert_a @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ).
% insert_filter_set_false
thf(fact_101_insert__filter__set__false,axiom,
! [P: nat > $o,X3: nat,A2: set_nat] :
( ~ ( P @ X3 )
=> ( ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ ( insert_nat @ X3 @ A2 ) )
& ( P @ A4 ) ) )
= ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( P @ A4 ) ) ) ) ) ).
% insert_filter_set_false
thf(fact_102_local_Odbl__def,axiom,
! [X3: a] :
( ( neg_dbl_a @ plus @ X3 )
= ( plus @ X3 @ X3 ) ) ).
% local.dbl_def
thf(fact_103_local_Osum__list_Ocomm__monoid__list__axioms,axiom,
groups1759187151362946711list_a @ plus @ zero ).
% local.sum_list.comm_monoid_list_axioms
thf(fact_104_local_Oprod__list_Ocomm__monoid__list__axioms,axiom,
groups1759187151362946711list_a @ times @ one2 ).
% local.prod_list.comm_monoid_list_axioms
thf(fact_105_local_ONats__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( semiring_Nats_a @ one2 @ plus @ zero ) )
=> ( ( member_a @ B @ ( semiring_Nats_a @ one2 @ plus @ zero ) )
=> ( member_a @ ( times @ A @ B ) @ ( semiring_Nats_a @ one2 @ plus @ zero ) ) ) ) ).
% local.Nats_mult
thf(fact_106_local_ONats__0,axiom,
member_a @ zero @ ( semiring_Nats_a @ one2 @ plus @ zero ) ).
% local.Nats_0
thf(fact_107_local_ONats__1,axiom,
member_a @ one2 @ ( semiring_Nats_a @ one2 @ plus @ zero ) ).
% local.Nats_1
thf(fact_108_local_ONats__add,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( semiring_Nats_a @ one2 @ plus @ zero ) )
=> ( ( member_a @ B @ ( semiring_Nats_a @ one2 @ plus @ zero ) )
=> ( member_a @ ( plus @ A @ B ) @ ( semiring_Nats_a @ one2 @ plus @ zero ) ) ) ) ).
% local.Nats_add
thf(fact_109_local_Osum__list_Omonoid__list__axioms,axiom,
groups_monoid_list_a @ plus @ zero ).
% local.sum_list.monoid_list_axioms
thf(fact_110_local_Oprod__list_Omonoid__list__axioms,axiom,
groups_monoid_list_a @ times @ one2 ).
% local.prod_list.monoid_list_axioms
thf(fact_111_local_Oof__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_neq_of_bool_a @ one2 @ zero
@ ( P
& Q ) )
= ( times @ ( zero_neq_of_bool_a @ one2 @ zero @ P ) @ ( zero_neq_of_bool_a @ one2 @ zero @ Q ) ) ) ).
% local.of_bool_conj
thf(fact_112_insert__Diff__single,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( insert_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_113_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_114_local_Oadd_Omonoid__axioms,axiom,
monoid_a @ plus @ zero ).
% local.add.monoid_axioms
thf(fact_115_local_Osplit__of__bool__asm,axiom,
! [P: a > $o,P2: $o] :
( ( P @ ( zero_neq_of_bool_a @ one2 @ zero @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one2 ) )
| ( ~ P2
& ~ ( P @ zero ) ) ) ) ) ).
% local.split_of_bool_asm
thf(fact_116_local_Osplit__of__bool,axiom,
! [P: a > $o,P2: $o] :
( ( P @ ( zero_neq_of_bool_a @ one2 @ zero @ P2 ) )
= ( ( P2
=> ( P @ one2 ) )
& ( ~ P2
=> ( P @ zero ) ) ) ) ).
% local.split_of_bool
thf(fact_117_local_Oof__bool__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ( zero_neq_of_bool_a @ one2 @ zero @ P2 )
= ( zero_neq_of_bool_a @ one2 @ zero @ Q2 ) )
= ( P2 = Q2 ) ) ).
% local.of_bool_eq_iff
thf(fact_118_local_Oof__bool__def,axiom,
! [P2: $o] :
( ( P2
=> ( ( zero_neq_of_bool_a @ one2 @ zero @ P2 )
= one2 ) )
& ( ~ P2
=> ( ( zero_neq_of_bool_a @ one2 @ zero @ P2 )
= zero ) ) ) ).
% local.of_bool_def
thf(fact_119_local_Oprod_Ofinite__Collect__op,axiom,
! [I3: set_a,X3: a > a,Y2: a > a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( X3 @ I2 )
!= one2 ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= one2 ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( times @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= one2 ) ) ) ) ) ) ).
% local.prod.finite_Collect_op
thf(fact_120_local_Oprod_Ofinite__Collect__op,axiom,
! [I3: set_b,X3: b > a,Y2: b > a] :
( ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( X3 @ I2 )
!= one2 ) ) ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= one2 ) ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( times @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= one2 ) ) ) ) ) ) ).
% local.prod.finite_Collect_op
thf(fact_121_local_Oprod_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > a,Y2: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( X3 @ I2 )
!= one2 ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= one2 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( times @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= one2 ) ) ) ) ) ) ).
% local.prod.finite_Collect_op
thf(fact_122_local_Osum_Ofinite__Collect__op,axiom,
! [I3: set_a,X3: a > a,Y2: a > a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( X3 @ I2 )
!= zero ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= zero ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ I3 )
& ( ( plus @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= zero ) ) ) ) ) ) ).
% local.sum.finite_Collect_op
thf(fact_123_local_Osum_Ofinite__Collect__op,axiom,
! [I3: set_b,X3: b > a,Y2: b > a] :
( ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( X3 @ I2 )
!= zero ) ) ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= zero ) ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [I2: b] :
( ( member_b @ I2 @ I3 )
& ( ( plus @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= zero ) ) ) ) ) ) ).
% local.sum.finite_Collect_op
thf(fact_124_local_Osum_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > a,Y2: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( X3 @ I2 )
!= zero ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( Y2 @ I2 )
!= zero ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I3 )
& ( ( plus @ ( X3 @ I2 ) @ ( Y2 @ I2 ) )
!= zero ) ) ) ) ) ) ).
% local.sum.finite_Collect_op
thf(fact_125_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X4: b] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_126_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_127_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X4: b] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_128_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_129_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_130_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X4: b] :
~ ( member_b @ X4 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_131_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_132_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_133_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_134_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_135_insert__absorb2,axiom,
! [X3: b,A2: set_b] :
( ( insert_b @ X3 @ ( insert_b @ X3 @ A2 ) )
= ( insert_b @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_136_insert__absorb2,axiom,
! [X3: nat,A2: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
= ( insert_nat @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_137_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_138_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_139_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_140_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_141_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_142_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_143_Diff__idemp,axiom,
! [A2: set_b,B2: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_144_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_145_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_146_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_147_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_148_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_149_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_150_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_151_local_Osum_Orelated,axiom,
! [R2: a > a > $o,S: set_b,H: b > a,G: b > a] :
( ( R2 @ zero @ zero )
=> ( ! [X1: a,Y1: a,X22: a,Y22: a] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus @ X1 @ Y1 ) @ ( plus @ X22 @ Y22 ) ) )
=> ( ( finite_finite_b @ S )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( R2 @ ( H @ X ) @ ( G @ X ) ) )
=> ( R2 @ ( groups1779759026887736869um_a_b @ plus @ zero @ H @ S ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S ) ) ) ) ) ) ).
% local.sum.related
thf(fact_152_local_Osum_Orelated,axiom,
! [R2: a > a > $o,S: set_nat,H: nat > a,G: nat > a] :
( ( R2 @ zero @ zero )
=> ( ! [X1: a,Y1: a,X22: a,Y22: a] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus @ X1 @ Y1 ) @ ( plus @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R2 @ ( H @ X ) @ ( G @ X ) ) )
=> ( R2 @ ( groups5773243554134465322_a_nat @ plus @ zero @ H @ S ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S ) ) ) ) ) ) ).
% local.sum.related
thf(fact_153_local_Omult_Omonoid__axioms,axiom,
monoid_a @ times @ one2 ).
% local.mult.monoid_axioms
thf(fact_154_local_Osum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_a,G: a > a > a,R2: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] :
( groups1779759026887736868um_a_a @ plus @ zero @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [Y3: a] :
( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_155_local_Osum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_b,G: a > b > a,R2: a > b > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] :
( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [Y3: b] :
( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_156_local_Osum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_nat,G: a > nat > a,R2: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] :
( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [Y3: nat] :
( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_157_local_Osum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_a,G: b > a > a,R2: b > a > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] :
( groups1779759026887736868um_a_a @ plus @ zero @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [Y3: a] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_158_local_Osum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_b,G: b > b > a,R2: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] :
( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [Y3: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_159_local_Osum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_nat,G: b > nat > a,R2: b > nat > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] :
( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [Y3: nat] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_160_local_Osum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_a,G: nat > a > a,R2: nat > a > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] :
( groups1779759026887736868um_a_a @ plus @ zero @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [Y3: a] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_161_local_Osum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_b,G: nat > b > a,R2: nat > b > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] :
( groups1779759026887736869um_a_b @ plus @ zero @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [Y3: b] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_162_local_Osum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_nat,G: nat > nat > a,R2: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] :
( groups5773243554134465322_a_nat @ plus @ zero @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [Y3: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.sum.swap_restrict
thf(fact_163_local_Osum_Ointer__filter,axiom,
! [A2: set_a,G: a > a,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [X4: a] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ zero )
@ A2 ) ) ) ).
% local.sum.inter_filter
thf(fact_164_local_Osum_Ointer__filter,axiom,
! [A2: set_b,G: b > a,P: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [X4: b] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ zero )
@ A2 ) ) ) ).
% local.sum.inter_filter
thf(fact_165_local_Osum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > a,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [X4: nat] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ zero )
@ A2 ) ) ) ).
% local.sum.inter_filter
thf(fact_166_local_Osum_Oinsert__if,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X3 @ A2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ ( insert_a @ X3 @ A2 ) )
= ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 ) ) )
& ( ~ ( member_a @ X3 @ A2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ ( insert_a @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 ) ) ) ) ) ) ).
% local.sum.insert_if
thf(fact_167_local_Osum_Oinsert__if,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X3 @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( insert_b @ X3 @ A2 ) )
= ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 ) ) )
& ( ~ ( member_b @ X3 @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( insert_b @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 ) ) ) ) ) ) ).
% local.sum.insert_if
thf(fact_168_local_Osum_Oinsert__if,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X3 @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( insert_nat @ X3 @ A2 ) )
= ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 ) ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( insert_nat @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 ) ) ) ) ) ) ).
% local.sum.insert_if
thf(fact_169_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_a,S: set_a,I: a > a,J: a > a,T: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_170_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_b,S: set_a,I: b > a,J: a > b,T: set_b,G: a > a,H: b > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_171_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_nat,S: set_a,I: nat > a,J: a > nat,T: set_nat,G: a > a,H: nat > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_172_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_a,S: set_b,I: a > b,J: b > a,T: set_a,G: b > a,H: a > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_173_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_b,S: set_b,I: b > b,J: b > b,T: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_174_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_nat,S: set_b,I: nat > b,J: b > nat,T: set_nat,G: b > a,H: nat > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_175_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_a,S: set_nat,I: a > nat,J: nat > a,T: set_a,G: nat > a,H: a > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_176_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_b,S: set_nat,I: b > nat,J: nat > b,T: set_b,G: nat > a,H: b > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_177_local_Osum_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_nat,S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= zero ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.sum.reindex_bij_witness_not_neutral
thf(fact_178_local_Osum_Osetdiff__irrelevant,axiom,
! [A2: set_b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G
@ ( minus_minus_set_b @ A2
@ ( collect_b
@ ^ [X4: b] :
( ( G @ X4 )
= zero ) ) ) )
= ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 ) ) ) ).
% local.sum.setdiff_irrelevant
thf(fact_179_local_Osum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= zero ) ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 ) ) ) ).
% local.sum.setdiff_irrelevant
thf(fact_180_local_Osum_Oremove,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X3 @ A2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736868um_a_a @ plus @ zero @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% local.sum.remove
thf(fact_181_local_Osum_Oremove,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X3 @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% local.sum.remove
thf(fact_182_local_Osum_Oremove,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= ( plus @ ( G @ X3 ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% local.sum.remove
thf(fact_183_local_Osum_Oinsert__remove,axiom,
! [A2: set_b,G: b > a,X3: b] :
( ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( insert_b @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ) ).
% local.sum.insert_remove
thf(fact_184_local_Osum_Oinsert__remove,axiom,
! [A2: set_nat,G: nat > a,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( insert_nat @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% local.sum.insert_remove
thf(fact_185_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_186_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_187_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_188_Diff__cancel,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ A2 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_189_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_190_empty__Diff,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_191_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_192_Diff__empty,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Diff_empty
thf(fact_193_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_194_local_Odouble__sum__split__case,axiom,
! [A2: set_b,F: b > b > a] :
( ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( F @ I2 ) @ A2 )
@ A2 )
= ( plus
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( F @ I2 @ I2 )
@ A2 )
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( groups1779759026887736869um_a_b @ plus @ zero @ ( F @ I2 ) @ ( minus_minus_set_b @ A2 @ ( insert_b @ I2 @ bot_bot_set_b ) ) )
@ A2 ) ) ) ) ).
% local.double_sum_split_case
thf(fact_195_local_Odouble__sum__split__case,axiom,
! [A2: set_nat,F: nat > nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( F @ I2 ) @ A2 )
@ A2 )
= ( plus
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( F @ I2 @ I2 )
@ A2 )
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( F @ I2 ) @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
@ A2 ) ) ) ) ).
% local.double_sum_split_case
thf(fact_196_local_Osum_Odelta__remove,axiom,
! [S: set_a,A: a,B: a > a,C: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus @ ( B @ A ) @ ( groups1779759026887736868um_a_a @ plus @ zero @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% local.sum.delta_remove
thf(fact_197_local_Osum_Odelta__remove,axiom,
! [S: set_b,A: b,B: b > a,C: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus @ ( B @ A ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% local.sum.delta_remove
thf(fact_198_local_Osum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > a,C: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus @ ( B @ A ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% local.sum.delta_remove
thf(fact_199_insert__Diff1,axiom,
! [X3: a,B2: set_a,A2: set_a] :
( ( member_a @ X3 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_200_insert__Diff1,axiom,
! [X3: b,B2: set_b,A2: set_b] :
( ( member_b @ X3 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_201_insert__Diff1,axiom,
! [X3: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_202_Diff__insert0,axiom,
! [X3: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X3 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_203_Diff__insert0,axiom,
! [X3: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X3 @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_204_Diff__insert0,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_205_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_206_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_207_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X4: b] : ( X4 = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_208_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X4: nat] : ( X4 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_209_local_Oof__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_neq_of_bool_a @ one2 @ zero @ P )
= one2 )
= P ) ).
% local.of_bool_eq_1_iff
thf(fact_210_local_Oof__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_neq_of_bool_a @ one2 @ zero @ P )
= zero )
= ~ P ) ).
% local.of_bool_eq_0_iff
thf(fact_211_local_Oof__bool__eq_I1_J,axiom,
( ( zero_neq_of_bool_a @ one2 @ zero @ $false )
= zero ) ).
% local.of_bool_eq(1)
thf(fact_212_local_Oof__bool__eq_I2_J,axiom,
( ( zero_neq_of_bool_a @ one2 @ zero @ $true )
= one2 ) ).
% local.of_bool_eq(2)
thf(fact_213_local_Odbl__simps_I2_J,axiom,
( ( neg_dbl_a @ plus @ zero )
= zero ) ).
% local.dbl_simps(2)
thf(fact_214_local_Osum_Oinfinite,axiom,
! [A2: set_b,G: b > a] :
( ~ ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= zero ) ) ).
% local.sum.infinite
thf(fact_215_local_Osum_Oinfinite,axiom,
! [A2: set_nat,G: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= zero ) ) ).
% local.sum.infinite
thf(fact_216_local_Osum_Odelta_H,axiom,
! [S: set_a,A: a,B: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta'
thf(fact_217_local_Osum_Odelta_H,axiom,
! [S: set_b,A: b,B: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta'
thf(fact_218_local_Osum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta'
thf(fact_219_local_Osum_Odelta,axiom,
! [S: set_a,A: a,B: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta
thf(fact_220_local_Osum_Odelta,axiom,
! [S: set_b,A: b,B: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta
thf(fact_221_local_Osum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ zero )
@ S )
= zero ) ) ) ) ).
% local.sum.delta
thf(fact_222_local_Odbl__inc__simps_I2_J,axiom,
( ( neg_dbl_inc_a @ plus @ one2 @ zero )
= one2 ) ).
% local.dbl_inc_simps(2)
thf(fact_223_local_Osum_Oinsert,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X3 @ A2 )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ ( insert_a @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 ) ) ) ) ) ).
% local.sum.insert
thf(fact_224_local_Osum_Oinsert,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X3 @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( insert_b @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 ) ) ) ) ) ).
% local.sum.insert
thf(fact_225_local_Osum_Oinsert,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X3 @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( insert_nat @ X3 @ A2 ) )
= ( plus @ ( G @ X3 ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 ) ) ) ) ) ).
% local.sum.insert
thf(fact_226_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_227_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X4: b] : ( member_b @ X4 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_228_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_229_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_230_equals0I,axiom,
! [A2: set_b] :
( ! [Y: b] :
~ ( member_b @ Y @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_231_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_232_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_233_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_234_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_235_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_236_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_237_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_238_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B4: set_b] :
( ( A2
= ( insert_b @ A @ B4 ) )
& ~ ( member_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_239_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_240_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_241_insert__commute,axiom,
! [X3: b,Y2: b,A2: set_b] :
( ( insert_b @ X3 @ ( insert_b @ Y2 @ A2 ) )
= ( insert_b @ Y2 @ ( insert_b @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_242_insert__commute,axiom,
! [X3: nat,Y2: nat,A2: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y2 @ A2 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_243_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_b] :
( ( A2
= ( insert_b @ B @ C3 ) )
& ~ ( member_b @ B @ C3 )
& ( B2
= ( insert_b @ A @ C3 ) )
& ~ ( member_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_244_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_245_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_246_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_247_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_248_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_249_insert__ident,axiom,
! [X3: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X3 @ A2 )
=> ( ~ ( member_b @ X3 @ B2 )
=> ( ( ( insert_b @ X3 @ A2 )
= ( insert_b @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_250_insert__ident,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( insert_nat @ X3 @ A2 )
= ( insert_nat @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_251_insert__ident,axiom,
! [X3: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X3 @ A2 )
=> ( ~ ( member_a @ X3 @ B2 )
=> ( ( ( insert_a @ X3 @ A2 )
= ( insert_a @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_252_Set_Oset__insert,axiom,
! [X3: b,A2: set_b] :
( ( member_b @ X3 @ A2 )
=> ~ ! [B4: set_b] :
( ( A2
= ( insert_b @ X3 @ B4 ) )
=> ( member_b @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_253_Set_Oset__insert,axiom,
! [X3: nat,A2: set_nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X3 @ B4 ) )
=> ( member_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_254_Set_Oset__insert,axiom,
! [X3: a,A2: set_a] :
( ( member_a @ X3 @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X3 @ B4 ) )
=> ( member_a @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_255_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_256_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_257_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_258_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_259_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_260_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_261_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_262_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_263_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_264_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_265_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_266_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_267_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_268_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_269_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_270_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_271_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_272_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_273_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X4: b] : $false ) ) ).
% empty_def
thf(fact_274_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X4: nat] : $false ) ) ).
% empty_def
thf(fact_275_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_276_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_277_insert__compr,axiom,
( insert_b
= ( ^ [A4: b,B5: set_b] :
( collect_b
@ ^ [X4: b] :
( ( X4 = A4 )
| ( member_b @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_278_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B5: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A4 )
| ( member_a @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_279_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B5: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( X4 = A4 )
| ( member_nat @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_280_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A5 )
& ~ ( member_a @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_281_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A5 )
& ~ ( member_b @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_282_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A5 )
& ~ ( member_nat @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_283_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_284_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_285_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_286_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_287_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_288_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_289_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_290_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_291_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_292_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_293_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_294_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_295_insert__Diff__if,axiom,
! [X3: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X3 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X3 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A2 ) @ B2 )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_296_insert__Diff__if,axiom,
! [X3: b,B2: set_b,A2: set_b] :
( ( ( member_b @ X3 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) )
& ( ~ ( member_b @ X3 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A2 ) @ B2 )
= ( insert_b @ X3 @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_297_insert__Diff__if,axiom,
! [X3: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_298_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X4: b] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X4: b] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_299_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_300_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X4: b] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X4: b] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_301_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_302_Diff__insert__absorb,axiom,
! [X3: a,A2: set_a] :
( ~ ( member_a @ X3 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A2 ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_303_Diff__insert__absorb,axiom,
! [X3: b,A2: set_b] :
( ~ ( member_b @ X3 @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A2 ) @ ( insert_b @ X3 @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_304_Diff__insert__absorb,axiom,
! [X3: nat,A2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_305_Diff__insert2,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_306_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_307_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_308_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_309_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_310_Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_311_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_312_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_313_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_314_local_Osum_OG__def,axiom,
! [I3: set_a,P2: a > a] :
( ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ I3 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ P2
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) ) ) )
& ( ~ ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ I3 )
= zero ) ) ) ).
% local.sum.G_def
thf(fact_315_local_Osum_OG__def,axiom,
! [I3: set_b,P2: b > a] :
( ( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ I3 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ P2
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) ) ) )
& ( ~ ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ I3 )
= zero ) ) ) ).
% local.sum.G_def
thf(fact_316_local_Osum_OG__def,axiom,
! [I3: set_nat,P2: nat > a] :
( ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ I3 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ P2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) ) ) )
& ( ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ I3 )
= zero ) ) ) ).
% local.sum.G_def
thf(fact_317_local_Osum__mset_Ocomm__monoid__mset__axioms,axiom,
comm_monoid_mset_a @ plus @ zero ).
% local.sum_mset.comm_monoid_mset_axioms
thf(fact_318_local_Oprod__mset_Ocomm__monoid__mset__axioms,axiom,
comm_monoid_mset_a @ times @ one2 ).
% local.prod_mset.comm_monoid_mset_axioms
thf(fact_319_local_Onot__iszero__1,axiom,
~ ( ring_iszero_a @ zero @ one2 ) ).
% local.not_iszero_1
thf(fact_320_local_Osum_Odistrib__triv_H,axiom,
! [I3: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ I3 )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero
@ ^ [I2: b] : ( plus @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( plus @ ( groups8906383913375152974um_a_b @ plus @ zero @ G @ I3 ) @ ( groups8906383913375152974um_a_b @ plus @ zero @ H @ I3 ) ) ) ) ).
% local.sum.distrib_triv'
thf(fact_321_local_Osum_Odistrib__triv_H,axiom,
! [I3: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ I3 )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero
@ ^ [I2: nat] : ( plus @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( plus @ ( groups6420517015690499521_a_nat @ plus @ zero @ G @ I3 ) @ ( groups6420517015690499521_a_nat @ plus @ zero @ H @ I3 ) ) ) ) ).
% local.sum.distrib_triv'
thf(fact_322_local_Osum_Odistrib_H,axiom,
! [I3: set_a,G: a > a,H: a > a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( G @ X4 )
!= zero ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( H @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero
@ ^ [I2: a] : ( plus @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( plus @ ( groups8906383913375152973um_a_a @ plus @ zero @ G @ I3 ) @ ( groups8906383913375152973um_a_a @ plus @ zero @ H @ I3 ) ) ) ) ) ).
% local.sum.distrib'
thf(fact_323_local_Osum_Odistrib_H,axiom,
! [I3: set_b,G: b > a,H: b > a] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( G @ X4 )
!= zero ) ) ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( H @ X4 )
!= zero ) ) ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero
@ ^ [I2: b] : ( plus @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( plus @ ( groups8906383913375152974um_a_b @ plus @ zero @ G @ I3 ) @ ( groups8906383913375152974um_a_b @ plus @ zero @ H @ I3 ) ) ) ) ) ).
% local.sum.distrib'
thf(fact_324_local_Osum_Odistrib_H,axiom,
! [I3: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( G @ X4 )
!= zero ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( H @ X4 )
!= zero ) ) ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero
@ ^ [I2: nat] : ( plus @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( plus @ ( groups6420517015690499521_a_nat @ plus @ zero @ G @ I3 ) @ ( groups6420517015690499521_a_nat @ plus @ zero @ H @ I3 ) ) ) ) ) ).
% local.sum.distrib'
thf(fact_325_local_Oprod_Odistrib__triv_H,axiom,
! [I3: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ I3 )
=> ( ( groups4667919067926330667od_a_b @ times @ one2
@ ^ [I2: b] : ( times @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( times @ ( groups4667919067926330667od_a_b @ times @ one2 @ G @ I3 ) @ ( groups4667919067926330667od_a_b @ times @ one2 @ H @ I3 ) ) ) ) ).
% local.prod.distrib_triv'
thf(fact_326_local_Oprod_Odistrib__triv_H,axiom,
! [I3: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ I3 )
=> ( ( groups7824906719281202852_a_nat @ times @ one2
@ ^ [I2: nat] : ( times @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( times @ ( groups7824906719281202852_a_nat @ times @ one2 @ G @ I3 ) @ ( groups7824906719281202852_a_nat @ times @ one2 @ H @ I3 ) ) ) ) ).
% local.prod.distrib_triv'
thf(fact_327_local_Oprod_Odistrib_H,axiom,
! [I3: set_a,G: a > a,H: a > a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( G @ X4 )
!= one2 ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( H @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2
@ ^ [I2: a] : ( times @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( times @ ( groups4667919067926330666od_a_a @ times @ one2 @ G @ I3 ) @ ( groups4667919067926330666od_a_a @ times @ one2 @ H @ I3 ) ) ) ) ) ).
% local.prod.distrib'
thf(fact_328_local_Oprod_Odistrib_H,axiom,
! [I3: set_b,G: b > a,H: b > a] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( G @ X4 )
!= one2 ) ) ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( H @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2
@ ^ [I2: b] : ( times @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( times @ ( groups4667919067926330667od_a_b @ times @ one2 @ G @ I3 ) @ ( groups4667919067926330667od_a_b @ times @ one2 @ H @ I3 ) ) ) ) ) ).
% local.prod.distrib'
thf(fact_329_local_Oprod_Odistrib_H,axiom,
! [I3: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( G @ X4 )
!= one2 ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( H @ X4 )
!= one2 ) ) ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2
@ ^ [I2: nat] : ( times @ ( G @ I2 ) @ ( H @ I2 ) )
@ I3 )
= ( times @ ( groups7824906719281202852_a_nat @ times @ one2 @ G @ I3 ) @ ( groups7824906719281202852_a_nat @ times @ one2 @ H @ I3 ) ) ) ) ) ).
% local.prod.distrib'
thf(fact_330_local_Opower__even__eq,axiom,
! [A: a,N: nat] :
( ( power_a @ one2 @ times @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_a @ one2 @ times @ ( power_a @ one2 @ times @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% local.power_even_eq
thf(fact_331_local_Opower__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_a @ one2 @ times @ zero @ N )
= one2 ) )
& ( ( N != zero_zero_nat )
=> ( ( power_a @ one2 @ times @ zero @ N )
= zero ) ) ) ).
% local.power_0_left
thf(fact_332_local_Oiszero__0,axiom,
ring_iszero_a @ zero @ zero ).
% local.iszero_0
thf(fact_333_local_Oiszero__def,axiom,
! [Z: a] :
( ( ring_iszero_a @ zero @ Z )
= ( Z = zero ) ) ).
% local.iszero_def
thf(fact_334_local_Oof__nat__aux_Osimps_I1_J,axiom,
! [Inc: a > a,I: a] :
( ( semiri4123672225490436309_aux_a @ Inc @ zero_zero_nat @ I )
= I ) ).
% local.of_nat_aux.simps(1)
thf(fact_335_local_Oprod_Ocong_H,axiom,
! [A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( A2 = B2 )
=> ( ! [X: a] :
( ( member_a @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ G @ A2 )
= ( groups4667919067926330666od_a_a @ times @ one2 @ H @ B2 ) ) ) ) ).
% local.prod.cong'
thf(fact_336_local_Osum_Ocong_H,axiom,
! [A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( A2 = B2 )
=> ( ! [X: a] :
( ( member_a @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ G @ A2 )
= ( groups8906383913375152973um_a_a @ plus @ zero @ H @ B2 ) ) ) ) ).
% local.sum.cong'
thf(fact_337_local_Oprod_Onon__neutral_H,axiom,
! [G: a > a,I3: set_a] :
( ( groups4667919067926330666od_a_a @ times @ one2 @ G
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( G @ X4 )
!= one2 ) ) ) )
= ( groups4667919067926330666od_a_a @ times @ one2 @ G @ I3 ) ) ).
% local.prod.non_neutral'
thf(fact_338_local_Oprod_Onon__neutral_H,axiom,
! [G: nat > a,I3: set_nat] :
( ( groups7824906719281202852_a_nat @ times @ one2 @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( G @ X4 )
!= one2 ) ) ) )
= ( groups7824906719281202852_a_nat @ times @ one2 @ G @ I3 ) ) ).
% local.prod.non_neutral'
thf(fact_339_local_Osum_Onon__neutral_H,axiom,
! [G: a > a,I3: set_a] :
( ( groups8906383913375152973um_a_a @ plus @ zero @ G
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( G @ X4 )
!= zero ) ) ) )
= ( groups8906383913375152973um_a_a @ plus @ zero @ G @ I3 ) ) ).
% local.sum.non_neutral'
thf(fact_340_local_Osum_Onon__neutral_H,axiom,
! [G: nat > a,I3: set_nat] :
( ( groups6420517015690499521_a_nat @ plus @ zero @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( G @ X4 )
!= zero ) ) ) )
= ( groups6420517015690499521_a_nat @ plus @ zero @ G @ I3 ) ) ).
% local.sum.non_neutral'
thf(fact_341_local_Opower_Opower__0,axiom,
! [A: a] :
( ( power_a @ one2 @ times @ A @ zero_zero_nat )
= one2 ) ).
% local.power.power_0
thf(fact_342_local_Opower__mult,axiom,
! [A: a,M: nat,N: nat] :
( ( power_a @ one2 @ times @ A @ ( times_times_nat @ M @ N ) )
= ( power_a @ one2 @ times @ ( power_a @ one2 @ times @ A @ M ) @ N ) ) ).
% local.power_mult
thf(fact_343_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_344_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_345_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_346_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_347_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_348_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_349_finite__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_350_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_351_finite__Diff2,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_352_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_353_local_Oprod_Oempty_H,axiom,
! [P2: b > a] :
( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ bot_bot_set_b )
= one2 ) ).
% local.prod.empty'
thf(fact_354_local_Oprod_Oempty_H,axiom,
! [P2: nat > a] :
( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ bot_bot_set_nat )
= one2 ) ).
% local.prod.empty'
thf(fact_355_local_Osum_Oempty_H,axiom,
! [P2: b > a] :
( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ bot_bot_set_b )
= zero ) ).
% local.sum.empty'
thf(fact_356_local_Osum_Oempty_H,axiom,
! [P2: nat > a] :
( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ bot_bot_set_nat )
= zero ) ).
% local.sum.empty'
thf(fact_357_local_Osum_Oeq__sum,axiom,
! [I3: set_b,P2: b > a] :
( ( finite_finite_b @ I3 )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ I3 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ P2 @ I3 ) ) ) ).
% local.sum.eq_sum
thf(fact_358_local_Osum_Oeq__sum,axiom,
! [I3: set_nat,P2: nat > a] :
( ( finite_finite_nat @ I3 )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ I3 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ P2 @ I3 ) ) ) ).
% local.sum.eq_sum
thf(fact_359_local_Oprod_Oinsert_H,axiom,
! [I3: set_a,P2: a > a,I: a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( ( member_a @ I @ I3 )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ ( insert_a @ I @ I3 ) )
= ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ I3 ) ) )
& ( ~ ( member_a @ I @ I3 )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ ( insert_a @ I @ I3 ) )
= ( times @ ( P2 @ I ) @ ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ I3 ) ) ) ) ) ) ).
% local.prod.insert'
thf(fact_360_local_Oprod_Oinsert_H,axiom,
! [I3: set_b,P2: b > a,I: b] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( ( member_b @ I @ I3 )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ ( insert_b @ I @ I3 ) )
= ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ I3 ) ) )
& ( ~ ( member_b @ I @ I3 )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ ( insert_b @ I @ I3 ) )
= ( times @ ( P2 @ I ) @ ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ I3 ) ) ) ) ) ) ).
% local.prod.insert'
thf(fact_361_local_Oprod_Oinsert_H,axiom,
! [I3: set_nat,P2: nat > a,I: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( ( member_nat @ I @ I3 )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ ( insert_nat @ I @ I3 ) )
= ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ I3 ) ) )
& ( ~ ( member_nat @ I @ I3 )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ ( insert_nat @ I @ I3 ) )
= ( times @ ( P2 @ I ) @ ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ I3 ) ) ) ) ) ) ).
% local.prod.insert'
thf(fact_362_local_Osum_Oinsert_H,axiom,
! [I3: set_a,P2: a > a,I: a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( ( member_a @ I @ I3 )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ ( insert_a @ I @ I3 ) )
= ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ I3 ) ) )
& ( ~ ( member_a @ I @ I3 )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ ( insert_a @ I @ I3 ) )
= ( plus @ ( P2 @ I ) @ ( groups8906383913375152973um_a_a @ plus @ zero @ P2 @ I3 ) ) ) ) ) ) ).
% local.sum.insert'
thf(fact_363_local_Osum_Oinsert_H,axiom,
! [I3: set_b,P2: b > a,I: b] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( ( member_b @ I @ I3 )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ ( insert_b @ I @ I3 ) )
= ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ I3 ) ) )
& ( ~ ( member_b @ I @ I3 )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ ( insert_b @ I @ I3 ) )
= ( plus @ ( P2 @ I ) @ ( groups8906383913375152974um_a_b @ plus @ zero @ P2 @ I3 ) ) ) ) ) ) ).
% local.sum.insert'
thf(fact_364_local_Osum_Oinsert_H,axiom,
! [I3: set_nat,P2: nat > a,I: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= zero ) ) ) )
=> ( ( ( member_nat @ I @ I3 )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ ( insert_nat @ I @ I3 ) )
= ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ I3 ) ) )
& ( ~ ( member_nat @ I @ I3 )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ ( insert_nat @ I @ I3 ) )
= ( plus @ ( P2 @ I ) @ ( groups6420517015690499521_a_nat @ plus @ zero @ P2 @ I3 ) ) ) ) ) ) ).
% local.sum.insert'
thf(fact_365_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_b,R2: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: b] :
( ( member_b @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_366_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_367_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_b,R2: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: b] :
( ( member_b @ X @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_368_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_nat,R2: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_369_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_b,R2: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: b] :
( ( member_b @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_370_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_371_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_372_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_373_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_374_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_375_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_376_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_377_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_378_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_379_Diff__infinite__finite,axiom,
! [T: set_b,S: set_b] :
( ( finite_finite_b @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_380_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_381_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A6: set_b] :
( ? [A3: b] :
( A
= ( insert_b @ A3 @ A6 ) )
=> ~ ( finite_finite_b @ A6 ) ) ) ) ).
% finite.cases
thf(fact_382_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_383_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A4: set_b] :
( ( A4 = bot_bot_set_b )
| ? [A5: set_b,B6: b] :
( ( A4
= ( insert_b @ B6 @ A5 ) )
& ( finite_finite_b @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_384_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A5: set_nat,B6: nat] :
( ( A4
= ( insert_nat @ B6 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_385_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_386_finite__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_387_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_388_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_389_finite__ne__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ! [X: b] : ( P @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ! [X: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_390_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_391_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_392_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A6: set_b] :
( ~ ( finite_finite_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_393_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_394_infinite__remove,axiom,
! [S: set_b,A: b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% infinite_remove
thf(fact_395_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_396_infinite__coinduct,axiom,
! [X5: set_b > $o,A2: set_b] :
( ( X5 @ A2 )
=> ( ! [A6: set_b] :
( ( X5 @ A6 )
=> ? [X2: b] :
( ( member_b @ X2 @ A6 )
& ( ( X5 @ ( minus_minus_set_b @ A6 @ ( insert_b @ X2 @ bot_bot_set_b ) ) )
| ~ ( finite_finite_b @ ( minus_minus_set_b @ A6 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) )
=> ~ ( finite_finite_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_397_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_398_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_399_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: b,A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( member_b @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_400_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_401_local_Opower__odd__eq,axiom,
! [A: a,N: nat] :
( ( power_a @ one2 @ times @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times @ A @ ( power_a @ one2 @ times @ ( power_a @ one2 @ times @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% local.power_odd_eq
thf(fact_402_local_Ozero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_a @ one2 @ times @ zero @ N )
= zero ) ) ).
% local.zero_power
thf(fact_403_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_404_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_405_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_406_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_407_local_Osum_Omono__neutral__cong__left,axiom,
! [T: set_a,S: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ X )
= zero ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ S )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left
thf(fact_408_local_Osum_Omono__neutral__cong__left,axiom,
! [T: set_b,S: set_b,H: b > a,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( H @ X )
= zero ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left
thf(fact_409_local_Osum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X )
= zero ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left
thf(fact_410_local_Oof__nat__aux_Osimps_I2_J,axiom,
! [Inc: a > a,N: nat,I: a] :
( ( semiri4123672225490436309_aux_a @ Inc @ ( suc @ N ) @ I )
= ( semiri4123672225490436309_aux_a @ Inc @ N @ ( Inc @ I ) ) ) ).
% local.of_nat_aux.simps(2)
thf(fact_411_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_412_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_413_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_414_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_415_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_416_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_417_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_418_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_419_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_420_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_421_local_Opower__Suc2,axiom,
! [A: a,N: nat] :
( ( power_a @ one2 @ times @ A @ ( suc @ N ) )
= ( times @ ( power_a @ one2 @ times @ A @ N ) @ A ) ) ).
% local.power_Suc2
thf(fact_422_local_Opower_Opower__Suc,axiom,
! [A: a,N: nat] :
( ( power_a @ one2 @ times @ A @ ( suc @ N ) )
= ( times @ A @ ( power_a @ one2 @ times @ A @ N ) ) ) ).
% local.power.power_Suc
thf(fact_423_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_424_local_Oprod_Omono__neutral__right_H,axiom,
! [S: set_b,T: set_b,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ G @ T )
= ( groups4667919067926330667od_a_b @ times @ one2 @ G @ S ) ) ) ) ).
% local.prod.mono_neutral_right'
thf(fact_425_local_Oprod_Omono__neutral__right_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ G @ T )
= ( groups7824906719281202852_a_nat @ times @ one2 @ G @ S ) ) ) ) ).
% local.prod.mono_neutral_right'
thf(fact_426_local_Oprod_Omono__neutral__left_H,axiom,
! [S: set_b,T: set_b,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ G @ S )
= ( groups4667919067926330667od_a_b @ times @ one2 @ G @ T ) ) ) ) ).
% local.prod.mono_neutral_left'
thf(fact_427_local_Oprod_Omono__neutral__left_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ G @ S )
= ( groups7824906719281202852_a_nat @ times @ one2 @ G @ T ) ) ) ) ).
% local.prod.mono_neutral_left'
thf(fact_428_local_Oprod_Omono__neutral__cong__right_H,axiom,
! [S: set_a,T: set_a,G: a > a,H: a > a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ G @ T )
= ( groups4667919067926330666od_a_a @ times @ one2 @ H @ S ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right'
thf(fact_429_local_Oprod_Omono__neutral__cong__right_H,axiom,
! [S: set_b,T: set_b,G: b > a,H: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ G @ T )
= ( groups4667919067926330667od_a_b @ times @ one2 @ H @ S ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right'
thf(fact_430_local_Oprod_Omono__neutral__cong__right_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a,H: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ G @ T )
= ( groups7824906719281202852_a_nat @ times @ one2 @ H @ S ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right'
thf(fact_431_local_Oprod_Omono__neutral__cong__left_H,axiom,
! [S: set_a,T: set_a,H: a > a,G: a > a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ! [I4: a] :
( ( member_a @ I4 @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ I4 )
= one2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ G @ S )
= ( groups4667919067926330666od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left'
thf(fact_432_local_Oprod_Omono__neutral__cong__left_H,axiom,
! [S: set_b,T: set_b,H: b > a,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [I4: b] :
( ( member_b @ I4 @ ( minus_minus_set_b @ T @ S ) )
=> ( ( H @ I4 )
= one2 ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ G @ S )
= ( groups4667919067926330667od_a_b @ times @ one2 @ H @ T ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left'
thf(fact_433_local_Oprod_Omono__neutral__cong__left_H,axiom,
! [S: set_nat,T: set_nat,H: nat > a,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ I4 )
= one2 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ G @ S )
= ( groups7824906719281202852_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left'
thf(fact_434_local_Osum_Omono__neutral__right_H,axiom,
! [S: set_b,T: set_b,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ G @ T )
= ( groups8906383913375152974um_a_b @ plus @ zero @ G @ S ) ) ) ) ).
% local.sum.mono_neutral_right'
thf(fact_435_local_Osum_Omono__neutral__right_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ G @ T )
= ( groups6420517015690499521_a_nat @ plus @ zero @ G @ S ) ) ) ) ).
% local.sum.mono_neutral_right'
thf(fact_436_local_Osum_Omono__neutral__left_H,axiom,
! [S: set_b,T: set_b,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ G @ S )
= ( groups8906383913375152974um_a_b @ plus @ zero @ G @ T ) ) ) ) ).
% local.sum.mono_neutral_left'
thf(fact_437_local_Osum_Omono__neutral__left_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ G @ S )
= ( groups6420517015690499521_a_nat @ plus @ zero @ G @ T ) ) ) ) ).
% local.sum.mono_neutral_left'
thf(fact_438_local_Osum_Omono__neutral__cong__right_H,axiom,
! [S: set_a,T: set_a,G: a > a,H: a > a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ G @ T )
= ( groups8906383913375152973um_a_a @ plus @ zero @ H @ S ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right'
thf(fact_439_local_Osum_Omono__neutral__cong__right_H,axiom,
! [S: set_b,T: set_b,G: b > a,H: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ G @ T )
= ( groups8906383913375152974um_a_b @ plus @ zero @ H @ S ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right'
thf(fact_440_local_Osum_Omono__neutral__cong__right_H,axiom,
! [S: set_nat,T: set_nat,G: nat > a,H: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ G @ T )
= ( groups6420517015690499521_a_nat @ plus @ zero @ H @ S ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right'
thf(fact_441_local_Osum_Omono__neutral__cong__left_H,axiom,
! [S: set_a,T: set_a,H: a > a,G: a > a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ! [I4: a] :
( ( member_a @ I4 @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ I4 )
= zero ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8906383913375152973um_a_a @ plus @ zero @ G @ S )
= ( groups8906383913375152973um_a_a @ plus @ zero @ H @ T ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left'
thf(fact_442_local_Osum_Omono__neutral__cong__left_H,axiom,
! [S: set_b,T: set_b,H: b > a,G: b > a] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ! [I4: b] :
( ( member_b @ I4 @ ( minus_minus_set_b @ T @ S ) )
=> ( ( H @ I4 )
= zero ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8906383913375152974um_a_b @ plus @ zero @ G @ S )
= ( groups8906383913375152974um_a_b @ plus @ zero @ H @ T ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left'
thf(fact_443_local_Osum_Omono__neutral__cong__left_H,axiom,
! [S: set_nat,T: set_nat,H: nat > a,G: nat > a] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ I4 )
= zero ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6420517015690499521_a_nat @ plus @ zero @ G @ S )
= ( groups6420517015690499521_a_nat @ plus @ zero @ H @ T ) ) ) ) ) ).
% local.sum.mono_neutral_cong_left'
thf(fact_444_local_Osum_Osubset__diff,axiom,
! [B2: set_b,A2: set_b,G: b > a] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( finite_finite_b @ A2 )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= ( plus @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ ( minus_minus_set_b @ A2 @ B2 ) ) @ ( groups1779759026887736869um_a_b @ plus @ zero @ G @ B2 ) ) ) ) ) ).
% local.sum.subset_diff
thf(fact_445_local_Osum_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ B2 ) ) ) ) ) ).
% local.sum.subset_diff
thf(fact_446_local_Osum_Osame__carrierI,axiom,
! [C2: set_a,A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ C2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ C2 ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrierI
thf(fact_447_local_Osum_Osame__carrierI,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ C2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ C2 ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrierI
thf(fact_448_local_Osum_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ C2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ C2 ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrierI
thf(fact_449_local_Osum_Osame__carrier,axiom,
! [C2: set_a,A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ A2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ B2 ) )
= ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ C2 )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrier
thf(fact_450_local_Osum_Osame__carrier,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ A2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ B2 ) )
= ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ C2 )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrier
thf(fact_451_local_Osum_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A3 )
= zero ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero ) )
=> ( ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ A2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ B2 ) )
= ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ C2 )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.sum.same_carrier
thf(fact_452_local_Osum_Omono__neutral__right,axiom,
! [T: set_b,S: set_b,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ T )
= ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S ) ) ) ) ) ).
% local.sum.mono_neutral_right
thf(fact_453_local_Osum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ T )
= ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S ) ) ) ) ) ).
% local.sum.mono_neutral_right
thf(fact_454_local_Osum_Omono__neutral__left,axiom,
! [T: set_b,S: set_b,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ S )
= ( groups1779759026887736869um_a_b @ plus @ zero @ G @ T ) ) ) ) ) ).
% local.sum.mono_neutral_left
thf(fact_455_local_Osum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ S )
= ( groups5773243554134465322_a_nat @ plus @ zero @ G @ T ) ) ) ) ) ).
% local.sum.mono_neutral_left
thf(fact_456_local_Osum_Omono__neutral__cong__right,axiom,
! [T: set_a,S: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736868um_a_a @ plus @ zero @ G @ T )
= ( groups1779759026887736868um_a_a @ plus @ zero @ H @ S ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right
thf(fact_457_local_Osum_Omono__neutral__cong__right,axiom,
! [T: set_b,S: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1779759026887736869um_a_b @ plus @ zero @ G @ T )
= ( groups1779759026887736869um_a_b @ plus @ zero @ H @ S ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right
thf(fact_458_local_Osum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ T )
= ( groups5773243554134465322_a_nat @ plus @ zero @ H @ S ) ) ) ) ) ) ).
% local.sum.mono_neutral_cong_right
thf(fact_459_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_460_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_461_semiring__numeral__class_Onumeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% semiring_numeral_class.numeral_times_numeral
thf(fact_462_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_463_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_464_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_465_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_466_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_467_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_468_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_469_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_470_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_471_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_472_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_473_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_474_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_475_insert__subset,axiom,
! [X3: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X3 @ A2 ) @ B2 )
= ( ( member_b @ X3 @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_476_insert__subset,axiom,
! [X3: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A2 ) @ B2 )
= ( ( member_a @ X3 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_477_insert__subset,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
= ( ( member_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_478_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B5: set_b] : ( ord_less_eq_set_b @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_479_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_480_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_481_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_482_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_483_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_484_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_485_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_486_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_487_Diff__eq__empty__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( minus_minus_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_488_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_489_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_490_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_491_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_492_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_493_local_Opower__Suc0__right,axiom,
! [A: a] :
( ( power_a @ one2 @ times @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% local.power_Suc0_right
thf(fact_494_local_Opower__0__Suc,axiom,
! [N: nat] :
( ( power_a @ one2 @ times @ zero @ ( suc @ N ) )
= zero ) ).
% local.power_0_Suc
thf(fact_495_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_496_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_497_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_498_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_499_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_500_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_501_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_502_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_503_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_504_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_505_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_506_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_507_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_508_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_509_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_510_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_511_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_512_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_513_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_514_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_515_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ X @ Y )
=> ( P @ ( suc @ X ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_516_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_517_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_518_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_519_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_520_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_521_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_522_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_523_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_524_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_525_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_526_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_527_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_528_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_529_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_530_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_531_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_532_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_533_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_534_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_535_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_536_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_537_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_538_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_539_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_540_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_541_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_542_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_543_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_544_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_545_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_546_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_547_Suc__inject,axiom,
! [X3: nat,Y2: nat] :
( ( ( suc @ X3 )
= ( suc @ Y2 ) )
=> ( X3 = Y2 ) ) ).
% Suc_inject
thf(fact_548_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_549_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_550_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_551_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_552_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_553_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_554_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_555_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_556_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_557_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_558_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_559_less__not__refl3,axiom,
! [S3: nat,T3: nat] :
( ( ord_less_nat @ S3 @ T3 )
=> ( S3 != T3 ) ) ).
% less_not_refl3
thf(fact_560_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_561_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_562_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_563_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_564_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_565_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_566_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_567_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_568_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_569_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_570_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_571_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_572_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_573_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_574_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_575_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_576_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_577_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_578_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A5 )
=> ( member_a @ T4 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_579_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A5 )
=> ( member_nat @ T4 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_580_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_581_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_582_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_583_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_584_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_585_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_586_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_587_in__mono,axiom,
! [A2: set_a,B2: set_a,X3: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_588_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_589_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_590_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_591_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_592_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_593_finite__has__maximal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X: num] :
( ( member_num @ X @ A2 )
& ( ord_less_eq_num @ A @ X )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_594_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ A @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_595_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_596_finite__has__minimal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X: num] :
( ( member_num @ X @ A2 )
& ( ord_less_eq_num @ X @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_597_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_598_finite__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_599_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_600_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_601_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_602_rev__finite__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_603_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_604_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_605_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_606_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_607_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_608_subset__insert,axiom,
! [X3: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X3 @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X3 @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_609_subset__insert,axiom,
! [X3: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X3 @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_610_subset__insert,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_611_insert__mono,axiom,
! [C2: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C2 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C2 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_612_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_613_double__diff,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ( minus_minus_set_b @ B2 @ ( minus_minus_set_b @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_614_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_615_Diff__subset,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_616_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_617_Diff__mono,axiom,
! [A2: set_b,C2: set_b,D2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ D2 @ B2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_618_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_619_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_620_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_621_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_622_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_623_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_624_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_625_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_626_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X4: b] : ( member_b @ X4 @ A5 )
@ ^ [X4: b] : ( member_b @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_627_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_628_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_629_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_630_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_631_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_632_finite__has__maximal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X: num] :
( ( member_num @ X @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_633_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_634_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_635_finite__has__minimal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X: num] :
( ( member_num @ X @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_636_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_637_subset__singleton__iff,axiom,
! [X5: set_b,A: b] :
( ( ord_less_eq_set_b @ X5 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X5 = bot_bot_set_b )
| ( X5
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_638_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_639_subset__singletonD,axiom,
! [A2: set_b,X3: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_640_subset__singletonD,axiom,
! [A2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_641_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X3: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_642_subset__Diff__insert,axiom,
! [A2: set_b,B2: set_b,X3: b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ ( insert_b @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C2 ) )
& ~ ( member_b @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_643_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X3: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_644_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_645_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_646_finite__subset__induct_H,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_647_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_648_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_649_finite__subset__induct,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_650_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_651_Diff__single__insert,axiom,
! [A2: set_b,X3: b,B2: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_652_Diff__single__insert,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_653_subset__insert__iff,axiom,
! [A2: set_a,X3: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X3 @ B2 ) )
= ( ( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_654_subset__insert__iff,axiom,
! [A2: set_b,X3: b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X3 @ B2 ) )
= ( ( ( member_b @ X3 @ A2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X3 @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_655_subset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_656_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_657_finite__remove__induct,axiom,
! [B2: set_b,P: set_b > $o] :
( ( finite_finite_b @ B2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( A6 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A6 @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_658_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_659_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_660_remove__induct,axiom,
! [P: set_b > $o,B2: set_b] :
( ( P @ bot_bot_set_b )
=> ( ( ~ ( finite_finite_b @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( A6 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A6 @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_661_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_662_ring__1_Oiszero_Ocong,axiom,
ring_iszero_a = ring_iszero_a ).
% ring_1.iszero.cong
thf(fact_663_semiring__1_ONats_Ocong,axiom,
semiring_Nats_a = semiring_Nats_a ).
% semiring_1.Nats.cong
thf(fact_664_neg__numeral_Odbl_Ocong,axiom,
neg_dbl_a = neg_dbl_a ).
% neg_numeral.dbl.cong
thf(fact_665_neg__numeral_Odbl__inc_Ocong,axiom,
neg_dbl_inc_a = neg_dbl_inc_a ).
% neg_numeral.dbl_inc.cong
thf(fact_666_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_667_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_668_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_669_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_670_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_671_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_672_local_Osum_Oin__pairs,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( plus @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.sum.in_pairs
thf(fact_673_local_Opower__minus__mult,axiom,
! [N: nat,A: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times @ ( power_a @ one2 @ times @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_a @ one2 @ times @ A @ N ) ) ) ).
% local.power_minus_mult
thf(fact_674_local_Osum_Oin__pairs__0,axiom,
! [G: nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( plus @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% local.sum.in_pairs_0
thf(fact_675_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_676_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_677_local_Osum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.sum.shift_bounds_cl_Suc_ivl
thf(fact_678_local_Osum_OatLeast0__atMost__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% local.sum.atLeast0_atMost_Suc
thf(fact_679_local_Osum__shift__lb__Suc0__0,axiom,
! [F: nat > a,K: nat] :
( ( ( F @ zero_zero_nat )
= zero )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups5773243554134465322_a_nat @ plus @ zero @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% local.sum_shift_lb_Suc0_0
thf(fact_680_local_Osum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus @ ( G @ M ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% local.sum.atLeast_Suc_atMost
thf(fact_681_local_Osum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus @ ( G @ ( suc @ N ) ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% local.sum.nat_ivl_Suc'
thf(fact_682_local_Opower__eq__if,axiom,
! [M: nat,P2: a] :
( ( ( M = zero_zero_nat )
=> ( ( power_a @ one2 @ times @ P2 @ M )
= one2 ) )
& ( ( M != zero_zero_nat )
=> ( ( power_a @ one2 @ times @ P2 @ M )
= ( times @ P2 @ ( power_a @ one2 @ times @ P2 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
% local.power_eq_if
thf(fact_683_local_Osum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus @ ( G @ M )
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% local.sum.Suc_reindex_ivl
thf(fact_684_local_Osum_OatMost__Suc__shift,axiom,
! [G: nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus @ ( G @ zero_zero_nat )
@ ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% local.sum.atMost_Suc_shift
thf(fact_685_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_686_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_687_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_688_cancel__comm__monoid__add__class_Odiff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% cancel_comm_monoid_add_class.diff_zero
thf(fact_689_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_690_monoid__mult__class_Omult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% monoid_mult_class.mult.right_neutral
thf(fact_691_Groups_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% Groups.mult_1
thf(fact_692_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_693_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_694_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_695_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_696_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_697_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_698_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_699_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_700_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_701_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_702_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_703_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_704_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_705_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_706_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_707_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_708_local_Osum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > a,H: nat > a] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] : ( if_a @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_a @ ( J2 = K ) @ zero @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] : ( if_a @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% local.sum.zero_middle
thf(fact_709_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_710_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_711_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_712_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_713_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_714_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_715_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_716_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_717_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_718_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_719_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_720_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_721_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_722_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_723_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_724_local_Opower__one__right,axiom,
! [A: a] :
( ( power_a @ one2 @ times @ A @ one_one_nat )
= A ) ).
% local.power_one_right
thf(fact_725_local_Osum_OatMost__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% local.sum.atMost_Suc
thf(fact_726_local_Osum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > a] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% local.sum.cl_ivl_Suc
thf(fact_727_comm__monoid__mult__class_Omult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% comm_monoid_mult_class.mult.comm_neutral
thf(fact_728_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_729_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_730_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_731_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_732_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_733_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_734_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_735_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_736_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_737_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_738_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_739_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_740_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_741_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_742_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_743_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_744_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_745_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_746_finite__psubset__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ! [B7: set_b] :
( ( ord_less_set_b @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_747_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_748_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_749_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_750_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_751_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_752_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_753_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_754_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_755_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_756_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_757_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_758_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_759_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_760_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_761_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_762_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_763_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R2 @ X @ X )
=> ( ! [X: nat,Y: nat,Z3: nat] :
( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_764_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_765_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_766_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_767_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_768_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_769_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_770_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_771_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_772_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_773_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_774_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_775_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_776_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_777_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_778_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_779_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_780_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_781_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_782_psubset__imp__ex__mem,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ? [B3: b] : ( member_b @ B3 @ ( minus_minus_set_b @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_783_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_784_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_785_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_786_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_787_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_788_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_789_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_790_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_791_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_792_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_793_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_794_monoid__mult__class_Omult_Omonoid__axioms,axiom,
monoid_nat @ times_times_nat @ one_one_nat ).
% monoid_mult_class.mult.monoid_axioms
thf(fact_795_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_796_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_797_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_798_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_799_numeral__class_Onumeral_Onumeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_class.numeral.numeral_One
thf(fact_800_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_801_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_802_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_803_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_804_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_805_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_806_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_807_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_808_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_809_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_810_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_811_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_812_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_813_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_814_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_815_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_816_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_817_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B6: nat] : ( times_times_nat @ B6 @ A4 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_818_semigroup__mult__class_Omult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_819_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_820_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_821_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_822_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_823_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_824_finite__induct__select,axiom,
! [S: set_b,P: set_b > $o] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [T5: set_b] :
( ( ord_less_set_b @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X2: b] :
( ( member_b @ X2 @ ( minus_minus_set_b @ S @ T5 ) )
& ( P @ ( insert_b @ X2 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_825_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ S @ T5 ) )
& ( P @ ( insert_nat @ X2 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_826_psubset__insert__iff,axiom,
! [A2: set_a,X3: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X3 @ B2 ) )
= ( ( ( member_a @ X3 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X3 @ B2 )
=> ( ( ( member_a @ X3 @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_827_psubset__insert__iff,axiom,
! [A2: set_b,X3: b,B2: set_b] :
( ( ord_less_set_b @ A2 @ ( insert_b @ X3 @ B2 ) )
= ( ( ( member_b @ X3 @ B2 )
=> ( ord_less_set_b @ A2 @ B2 ) )
& ( ~ ( member_b @ X3 @ B2 )
=> ( ( ( member_b @ X3 @ A2 )
=> ( ord_less_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X3 @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_828_psubset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_829_monoid_Oright__neutral,axiom,
! [F: a > a > a,Z: a,A: a] :
( ( monoid_a @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_830_monoid_Oleft__neutral,axiom,
! [F: a > a > a,Z: a,A: a] :
( ( monoid_a @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_831_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_832_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_833_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_834_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_835_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_836_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_837_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_838_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_839_Icc__subset__Iic__iff,axiom,
! [L2: num,H: num,H2: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L2 @ H ) @ ( set_ord_atMost_num @ H2 ) )
= ( ~ ( ord_less_eq_num @ L2 @ H )
| ( ord_less_eq_num @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_840_Icc__subset__Iic__iff,axiom,
! [L2: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L2 @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_841_Icc__subset__Iic__iff,axiom,
! [L2: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_842_atMost__subset__iff,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y2 ) )
= ( ord_less_eq_num @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_843_atMost__subset__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X3 ) @ ( set_or4236626031148496127et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_844_atMost__subset__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_845_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_846_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_847_atMost__eq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X3 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_848_atLeastAtMost__iff,axiom,
! [I: num,L2: num,U2: num] :
( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U2 ) )
= ( ( ord_less_eq_num @ L2 @ I )
& ( ord_less_eq_num @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_849_atLeastAtMost__iff,axiom,
! [I: set_nat,L2: set_nat,U2: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_set_nat @ L2 @ I )
& ( ord_less_eq_set_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_850_atLeastAtMost__iff,axiom,
! [I: nat,L2: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_nat @ L2 @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_851_Icc__eq__Icc,axiom,
! [L2: num,H: num,L3: num,H2: num] :
( ( ( set_or7049704709247886629st_num @ L2 @ H )
= ( set_or7049704709247886629st_num @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_num @ L2 @ H )
& ~ ( ord_less_eq_num @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_852_Icc__eq__Icc,axiom,
! [L2: set_nat,H: set_nat,L3: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L2 @ H )
= ( set_or4548717258645045905et_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L2 @ H )
& ~ ( ord_less_eq_set_nat @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_853_Icc__eq__Icc,axiom,
! [L2: nat,H: nat,L3: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L2 @ H )
= ( set_or1269000886237332187st_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L2 @ H )
& ~ ( ord_less_eq_nat @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_854_atMost__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
= ( ord_less_eq_num @ I @ K ) ) ).
% atMost_iff
thf(fact_855_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_856_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_857_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_858_finite__atLeastAtMost,axiom,
! [L2: nat,U2: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) ) ).
% finite_atLeastAtMost
thf(fact_859_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_860_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_861_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_862_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_863_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_864_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_865_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_866_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_867_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_868_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_869_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_870_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_871_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_872_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_873_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_874_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_875_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_876_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_877_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_878_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_879_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U: num] :
( collect_num
@ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_880_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_881_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_882_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_883_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_884_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D )
& ( ( ord_less_num @ C @ A )
| ( ord_less_num @ B @ D ) ) ) )
& ( ord_less_eq_num @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_885_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_886_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_887_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_888_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_889_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_890_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_891_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_892_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_893_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_894_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_895_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_896_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_897_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_898_mult__zero__class_Omult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_class.mult_zero_right
thf(fact_899_mult__zero__class_Omult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_class.mult_zero_left
thf(fact_900_zero__neq__one_Oof__bool_Ocong,axiom,
zero_neq_of_bool_a = zero_neq_of_bool_a ).
% zero_neq_one.of_bool.cong
thf(fact_901_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_902_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_903_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_904_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_905_mult__zero__class_Omult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_zero_class.mult_not_zero
thf(fact_906_zero__neq__one__class_Ozero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one_class.zero_neq_one
thf(fact_907_comm__semiring__1__cancel__class_Oleft__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% comm_semiring_1_cancel_class.left_diff_distrib'
thf(fact_908_comm__semiring__1__cancel__class_Oright__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% comm_semiring_1_cancel_class.right_diff_distrib'
thf(fact_909_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_910_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_911_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_912_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_913_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_914_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_915_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_916_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_917_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_918_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_919_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_920_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_921_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_922_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_923_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_924_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_925_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_926_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_927_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_928_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_929_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_930_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_931_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_932_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_933_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_934_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_935_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_936_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_937_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_938_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_939_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_940_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_941_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_942_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_943_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_944_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_945_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_946_local_Oprod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > a,H: nat > a] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [J2: nat] : ( if_a @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_a @ ( J2 = K ) @ one2 @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [J2: nat] : ( if_a @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% local.prod.zero_middle
thf(fact_947_local_Osum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > a,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% local.sum.ub_add_nat
thf(fact_948_local_Oprod_Oreindex__bij__witness,axiom,
! [S: set_a,I: a > a,J: a > a,T: set_a,H: a > a,G: a > a] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness
thf(fact_949_local_Oprod_Oreindex__bij__witness,axiom,
! [S: set_a,I: nat > a,J: a > nat,T: set_nat,H: nat > a,G: a > a] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness
thf(fact_950_local_Oprod_Oreindex__bij__witness,axiom,
! [S: set_nat,I: a > nat,J: nat > a,T: set_a,H: a > a,G: nat > a] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness
thf(fact_951_local_Oprod_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H: nat > a,G: nat > a] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness
thf(fact_952_local_Oprod_Onot__neutral__contains__not__neutral,axiom,
! [G: a > a,A2: set_a] :
( ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 )
!= one2 )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( G @ A3 )
= one2 ) ) ) ).
% local.prod.not_neutral_contains_not_neutral
thf(fact_953_local_Oprod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > a,A2: set_nat] :
( ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
!= one2 )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= one2 ) ) ) ).
% local.prod.not_neutral_contains_not_neutral
thf(fact_954_local_Oprod_Oneutral,axiom,
! [A2: set_nat,G: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= one2 ) ) ).
% local.prod.neutral
thf(fact_955_local_Oprod_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > a,A2: set_a,H: a > a,Gamma: a > a,Phi: a > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ( ( member_a @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ Phi @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general_inverses
thf(fact_956_local_Oprod_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > a,A2: set_a,H: a > nat,Gamma: nat > a,Phi: a > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( ( member_a @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ Phi @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general_inverses
thf(fact_957_local_Oprod_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > nat,A2: set_nat,H: nat > a,Gamma: a > a,Phi: nat > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ( ( member_nat @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ Phi @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general_inverses
thf(fact_958_local_Oprod_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( ( member_nat @ ( K @ Y ) @ A2 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ Phi @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general_inverses
thf(fact_959_local_Oprod_Oeq__general,axiom,
! [B2: set_a,A2: set_a,H: a > a,Gamma: a > a,Phi: a > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ Phi @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general
thf(fact_960_local_Oprod_Oeq__general,axiom,
! [B2: set_nat,A2: set_a,H: a > nat,Gamma: nat > a,Phi: a > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ Phi @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general
thf(fact_961_local_Oprod_Oeq__general,axiom,
! [B2: set_a,A2: set_nat,H: nat > a,Gamma: a > a,Phi: nat > a] :
( ! [Y: a] :
( ( member_a @ Y @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ Phi @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general
thf(fact_962_local_Oprod_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ Phi @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ Gamma @ B2 ) ) ) ) ).
% local.prod.eq_general
thf(fact_963_local_Oprod_Ocong,axiom,
! [A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( A2 = B2 )
=> ( ! [X: a] :
( ( member_a @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ B2 ) ) ) ) ).
% local.prod.cong
thf(fact_964_local_Oprod_Ocong,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ B2 ) ) ) ) ).
% local.prod.cong
thf(fact_965_local_Oprod_Oswap,axiom,
! [G: nat > nat > a,B2: set_nat,A2: set_nat] :
( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( groups1957776620359388557_a_nat @ times @ one2 @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [J2: nat] :
( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% local.prod.swap
thf(fact_966_local_Oprod_Odistrib,axiom,
! [G: nat > a,H: nat > a,A2: set_nat] :
( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] : ( times @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ H @ A2 ) ) ) ).
% local.prod.distrib
thf(fact_967_cancel__semigroup__add__class_Oadd__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% cancel_semigroup_add_class.add_right_cancel
thf(fact_968_cancel__semigroup__add__class_Oadd__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% cancel_semigroup_add_class.add_left_cancel
thf(fact_969_local_Oprod_Orelated,axiom,
! [R2: a > a > $o,S: set_b,H: b > a,G: b > a] :
( ( R2 @ one2 @ one2 )
=> ( ! [X1: a,Y1: a,X22: a,Y22: a] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( times @ X1 @ Y1 ) @ ( times @ X22 @ Y22 ) ) )
=> ( ( finite_finite_b @ S )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( R2 @ ( H @ X ) @ ( G @ X ) ) )
=> ( R2 @ ( groups2061451144089001602od_a_b @ times @ one2 @ H @ S ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S ) ) ) ) ) ) ).
% local.prod.related
thf(fact_970_local_Oprod_Orelated,axiom,
! [R2: a > a > $o,S: set_nat,H: nat > a,G: nat > a] :
( ( R2 @ one2 @ one2 )
=> ( ! [X1: a,Y1: a,X22: a,Y22: a] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( times @ X1 @ Y1 ) @ ( times @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R2 @ ( H @ X ) @ ( G @ X ) ) )
=> ( R2 @ ( groups1957776620359388557_a_nat @ times @ one2 @ H @ S ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S ) ) ) ) ) ) ).
% local.prod.related
thf(fact_971_local_Opower__add,axiom,
! [A: a,M: nat,N: nat] :
( ( power_a @ one2 @ times @ A @ ( plus_plus_nat @ M @ N ) )
= ( times @ ( power_a @ one2 @ times @ A @ M ) @ ( power_a @ one2 @ times @ A @ N ) ) ) ).
% local.power_add
thf(fact_972_local_Oprod_Oswap__restrict,axiom,
! [A2: set_a,B2: set_a,G: a > a > a,R2: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] :
( groups2061451144089001601od_a_a @ times @ one2 @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [Y3: a] :
( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_973_local_Oprod_Oswap__restrict,axiom,
! [A2: set_a,B2: set_b,G: a > b > a,R2: a > b > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] :
( groups2061451144089001602od_a_b @ times @ one2 @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [Y3: b] :
( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_974_local_Oprod_Oswap__restrict,axiom,
! [A2: set_b,B2: set_a,G: b > a > a,R2: b > a > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] :
( groups2061451144089001601od_a_a @ times @ one2 @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [Y3: a] :
( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_975_local_Oprod_Oswap__restrict,axiom,
! [A2: set_b,B2: set_b,G: b > b > a,R2: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] :
( groups2061451144089001602od_a_b @ times @ one2 @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [Y3: b] :
( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_976_local_Oprod_Oswap__restrict,axiom,
! [A2: set_a,B2: set_nat,G: a > nat > a,R2: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] :
( groups1957776620359388557_a_nat @ times @ one2 @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [Y3: nat] :
( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] : ( G @ X4 @ Y3 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_977_local_Oprod_Oswap__restrict,axiom,
! [A2: set_b,B2: set_nat,G: b > nat > a,R2: b > nat > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] :
( groups1957776620359388557_a_nat @ times @ one2 @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [Y3: nat] :
( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] : ( G @ X4 @ Y3 )
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_978_local_Oprod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_a,G: nat > a > a,R2: nat > a > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] :
( groups2061451144089001601od_a_a @ times @ one2 @ ( G @ X4 )
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [Y3: a] :
( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_979_local_Oprod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_b,G: nat > b > a,R2: nat > b > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] :
( groups2061451144089001602od_a_b @ times @ one2 @ ( G @ X4 )
@ ( collect_b
@ ^ [Y3: b] :
( ( member_b @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [Y3: b] :
( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_980_local_Oprod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_nat,G: nat > nat > a,R2: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] :
( groups1957776620359388557_a_nat @ times @ one2 @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [Y3: nat] :
( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( R2 @ X4 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% local.prod.swap_restrict
thf(fact_981_local_Oprod_Ointer__filter,axiom,
! [A2: set_a,G: a > a,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [X4: a] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ one2 )
@ A2 ) ) ) ).
% local.prod.inter_filter
thf(fact_982_local_Oprod_Ointer__filter,axiom,
! [A2: set_b,G: b > a,P: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [X4: b] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ one2 )
@ A2 ) ) ) ).
% local.prod.inter_filter
thf(fact_983_local_Oprod_Ointer__filter,axiom,
! [A2: set_nat,G: nat > a,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [X4: nat] : ( if_a @ ( P @ X4 ) @ ( G @ X4 ) @ one2 )
@ A2 ) ) ) ).
% local.prod.inter_filter
thf(fact_984_local_Oprod_Oinsert__if,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X3 @ A2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ ( insert_a @ X3 @ A2 ) )
= ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 ) ) )
& ( ~ ( member_a @ X3 @ A2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ ( insert_a @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 ) ) ) ) ) ) ).
% local.prod.insert_if
thf(fact_985_local_Oprod_Oinsert__if,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X3 @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( insert_b @ X3 @ A2 ) )
= ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 ) ) )
& ( ~ ( member_b @ X3 @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( insert_b @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 ) ) ) ) ) ) ).
% local.prod.insert_if
thf(fact_986_local_Oprod_Oinsert__if,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X3 @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( insert_nat @ X3 @ A2 ) )
= ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 ) ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( insert_nat @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 ) ) ) ) ) ) ).
% local.prod.insert_if
thf(fact_987_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_a,S: set_a,I: a > a,J: a > a,T: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_988_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_b,S: set_a,I: b > a,J: a > b,T: set_b,G: a > a,H: b > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_989_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_a,S: set_b,I: a > b,J: b > a,T: set_a,G: b > a,H: a > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_990_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_b,S: set_b,I: b > b,J: b > b,T: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_991_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_a,T2: set_nat,S: set_a,I: nat > a,J: a > nat,T: set_nat,G: a > a,H: nat > a] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_a @ ( I @ B3 ) @ ( minus_minus_set_a @ S @ S2 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_992_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_b,T2: set_nat,S: set_b,I: nat > b,J: b > nat,T: set_nat,G: b > a,H: nat > a] :
( ( finite_finite_b @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_b @ ( I @ B3 ) @ ( minus_minus_set_b @ S @ S2 ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_993_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_a,S: set_nat,I: a > nat,J: nat > a,T: set_a,G: nat > a,H: a > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T2 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_994_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_b,S: set_nat,I: b > nat,J: nat > b,T: set_b,G: nat > a,H: b > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_b @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_b @ ( J @ A3 ) @ ( minus_minus_set_b @ T @ T2 ) ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_995_local_Oprod_Oreindex__bij__witness__not__neutral,axiom,
! [S2: set_nat,T2: set_nat,S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S2 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T2 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T2 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S @ S2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( H @ B3 )
= one2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% local.prod.reindex_bij_witness_not_neutral
thf(fact_996_local_Oprod_Osetdiff__irrelevant,axiom,
! [A2: set_b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G
@ ( minus_minus_set_b @ A2
@ ( collect_b
@ ^ [X4: b] :
( ( G @ X4 )
= one2 ) ) ) )
= ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 ) ) ) ).
% local.prod.setdiff_irrelevant
thf(fact_997_local_Oprod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= one2 ) ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 ) ) ) ).
% local.prod.setdiff_irrelevant
thf(fact_998_local_Oprod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.prod.shift_bounds_cl_Suc_ivl
thf(fact_999_local_Oprod__zero,axiom,
! [A2: set_b,F: b > a] :
( ( finite_finite_b @ A2 )
=> ( ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( ( F @ X2 )
= zero ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ F @ A2 )
= zero ) ) ) ).
% local.prod_zero
thf(fact_1000_local_Oprod__zero,axiom,
! [A2: set_nat,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F @ X2 )
= zero ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ F @ A2 )
= zero ) ) ) ).
% local.prod_zero
thf(fact_1001_local_Osum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > a,M: nat,K: nat,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.sum.shift_bounds_cl_nat_ivl
thf(fact_1002_local_Oprod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > a,M: nat,K: nat,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.prod.shift_bounds_cl_nat_ivl
thf(fact_1003_local_Oprod_OG__def,axiom,
! [I3: set_a,P2: a > a] :
( ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ I3 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ P2
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) ) ) )
& ( ~ ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330666od_a_a @ times @ one2 @ P2 @ I3 )
= one2 ) ) ) ).
% local.prod.G_def
thf(fact_1004_local_Oprod_OG__def,axiom,
! [I3: set_b,P2: b > a] :
( ( ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ I3 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ P2
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) ) ) )
& ( ~ ( finite_finite_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ I3 )
= one2 ) ) ) ).
% local.prod.G_def
thf(fact_1005_local_Oprod_OG__def,axiom,
! [I3: set_nat,P2: nat > a] :
( ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ I3 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ P2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) ) ) )
& ( ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ I3 )
& ( ( P2 @ X4 )
!= one2 ) ) ) )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ I3 )
= one2 ) ) ) ).
% local.prod.G_def
thf(fact_1006_local_Oprod_Osubset__diff,axiom,
! [B2: set_b,A2: set_b,G: b > a] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( finite_finite_b @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 )
= ( times @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( minus_minus_set_b @ A2 @ B2 ) ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ B2 ) ) ) ) ) ).
% local.prod.subset_diff
thf(fact_1007_local_Oprod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G: nat > a] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ B2 ) ) ) ) ) ).
% local.prod.subset_diff
thf(fact_1008_local_Oprod_Osame__carrierI,axiom,
! [C2: set_a,A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ C2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ C2 ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrierI
thf(fact_1009_local_Oprod_Osame__carrierI,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ C2 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ C2 ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrierI
thf(fact_1010_local_Oprod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ C2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ C2 ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ B2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrierI
thf(fact_1011_local_Oprod_Osame__carrier,axiom,
! [C2: set_a,A2: set_a,B2: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ B2 ) )
= ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ C2 )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrier
thf(fact_1012_local_Oprod_Osame__carrier,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ B2 ) )
= ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ C2 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrier
thf(fact_1013_local_Oprod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A3 )
= one2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= one2 ) )
=> ( ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ B2 ) )
= ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ C2 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ C2 ) ) ) ) ) ) ) ) ).
% local.prod.same_carrier
thf(fact_1014_local_Oprod_Omono__neutral__right,axiom,
! [T: set_b,S: set_b,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ T )
= ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S ) ) ) ) ) ).
% local.prod.mono_neutral_right
thf(fact_1015_local_Oprod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ T )
= ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S ) ) ) ) ) ).
% local.prod.mono_neutral_right
thf(fact_1016_local_Oprod_Omono__neutral__left,axiom,
! [T: set_b,S: set_b,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ G @ T ) ) ) ) ) ).
% local.prod.mono_neutral_left
thf(fact_1017_local_Oprod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ G @ T ) ) ) ) ) ).
% local.prod.mono_neutral_left
thf(fact_1018_local_Oprod_Omono__neutral__cong__right,axiom,
! [T: set_a,S: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ T )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ S ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right
thf(fact_1019_local_Oprod_Omono__neutral__cong__right,axiom,
! [T: set_b,S: set_b,G: b > a,H: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ T )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ S ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right
thf(fact_1020_local_Oprod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= one2 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ T )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ S ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_right
thf(fact_1021_local_Oprod_Omono__neutral__cong__left,axiom,
! [T: set_a,S: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ X )
= one2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ H @ T ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left
thf(fact_1022_local_Oprod_Omono__neutral__cong__left,axiom,
! [T: set_b,S: set_b,H: b > a,G: b > a] :
( ( finite_finite_b @ T )
=> ( ( ord_less_eq_set_b @ S @ T )
=> ( ! [X: b] :
( ( member_b @ X @ ( minus_minus_set_b @ T @ S ) )
=> ( ( H @ X )
= one2 ) )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ H @ T ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left
thf(fact_1023_local_Oprod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X )
= one2 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ H @ T ) ) ) ) ) ) ).
% local.prod.mono_neutral_cong_left
thf(fact_1024_local_Oprod_OatLeast0__atMost__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% local.prod.atLeast0_atMost_Suc
thf(fact_1025_local_Oprod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times @ ( G @ ( suc @ N ) ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% local.prod.nat_ivl_Suc'
thf(fact_1026_local_Oprod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times @ ( G @ M ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% local.prod.atLeast_Suc_atMost
thf(fact_1027_local_Oprod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times @ ( G @ M )
@ ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% local.prod.Suc_reindex_ivl
thf(fact_1028_local_Oprod_OatMost__Suc__shift,axiom,
! [G: nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times @ ( G @ zero_zero_nat )
@ ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% local.prod.atMost_Suc_shift
thf(fact_1029_local_Osum_OatLeastAtMost__rev,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% local.sum.atLeastAtMost_rev
thf(fact_1030_local_Oprod_OatLeastAtMost__rev,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% local.prod.atLeastAtMost_rev
thf(fact_1031_local_Oprod_Oremove,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X3 @ A2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001601od_a_a @ times @ one2 @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% local.prod.remove
thf(fact_1032_local_Oprod_Oremove,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X3 @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% local.prod.remove
thf(fact_1033_local_Oprod_Oremove,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= ( times @ ( G @ X3 ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% local.prod.remove
thf(fact_1034_local_Oprod_Oinsert__remove,axiom,
! [A2: set_b,G: b > a,X3: b] :
( ( finite_finite_b @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( insert_b @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ) ).
% local.prod.insert_remove
thf(fact_1035_local_Oprod_Oinsert__remove,axiom,
! [A2: set_nat,G: nat > a,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( insert_nat @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% local.prod.insert_remove
thf(fact_1036_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1037_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1038_monoid__add__class_Oadd_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% monoid_add_class.add.right_neutral
thf(fact_1039_cancel__comm__monoid__add__class_Oadd__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% cancel_comm_monoid_add_class.add_cancel_left_left
thf(fact_1040_cancel__comm__monoid__add__class_Oadd__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% cancel_comm_monoid_add_class.add_cancel_left_right
thf(fact_1041_cancel__comm__monoid__add__class_Oadd__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% cancel_comm_monoid_add_class.add_cancel_right_left
thf(fact_1042_cancel__comm__monoid__add__class_Oadd__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% cancel_comm_monoid_add_class.add_cancel_right_right
thf(fact_1043_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1044_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y2 ) )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1045_Groups_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% Groups.add_0
thf(fact_1046_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1047_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1048_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1049_numeral__class_Onumeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_class.numeral_plus_numeral
thf(fact_1050_cancel__ab__semigroup__add__class_Oadd__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% cancel_ab_semigroup_add_class.add_diff_cancel_left
thf(fact_1051_cancel__ab__semigroup__add__class_Oadd__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% cancel_ab_semigroup_add_class.add_diff_cancel_left'
thf(fact_1052_cancel__ab__semigroup__add__class_Oadd__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% cancel_ab_semigroup_add_class.add_diff_cancel_right
thf(fact_1053_cancel__ab__semigroup__add__class_Oadd__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% cancel_ab_semigroup_add_class.add_diff_cancel_right'
thf(fact_1054_local_Oprod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > a,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% local.prod.ub_add_nat
thf(fact_1055_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1056_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1057_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1058_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1059_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1060_local_Oprod_Odelta__remove,axiom,
! [S: set_a,A: a,B: a > a,C: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( times @ ( B @ A ) @ ( groups2061451144089001601od_a_a @ times @ one2 @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups2061451144089001601od_a_a @ times @ one2 @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% local.prod.delta_remove
thf(fact_1061_local_Oprod_Odelta__remove,axiom,
! [S: set_b,A: b,B: b > a,C: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( times @ ( B @ A ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups2061451144089001602od_a_b @ times @ one2 @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% local.prod.delta_remove
thf(fact_1062_local_Oprod_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > a,C: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( times @ ( B @ A ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups1957776620359388557_a_nat @ times @ one2 @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% local.prod.delta_remove
thf(fact_1063_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1064_local_Oprod_Oin__pairs,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( times @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% local.prod.in_pairs
thf(fact_1065_local_Oprod_Oin__pairs__0,axiom,
! [G: nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( times @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% local.prod.in_pairs_0
thf(fact_1066_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1067_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1068_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1069_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1070_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1071_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1072_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1073_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1074_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1075_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1076_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1077_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1078_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1079_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1080_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1081_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1082_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1083_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1084_numeral__class_Onumeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_class.numeral_plus_one
thf(fact_1085_numeral__class_Oone__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% numeral_class.one_plus_numeral
thf(fact_1086_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1087_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1088_local_Oprod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [Uu: nat] : one2
@ A2 )
= one2 ) ).
% local.prod.neutral_const
thf(fact_1089_numeral__class_Oone__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% numeral_class.one_add_one
thf(fact_1090_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1091_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1092_local_Oprod_Oempty,axiom,
! [G: b > a] :
( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ bot_bot_set_b )
= one2 ) ).
% local.prod.empty
thf(fact_1093_local_Oprod_Oempty,axiom,
! [G: nat > a] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ bot_bot_set_nat )
= one2 ) ).
% local.prod.empty
thf(fact_1094_local_Oprod_Oinfinite,axiom,
! [A2: set_b,G: b > a] :
( ~ ( finite_finite_b @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 )
= one2 ) ) ).
% local.prod.infinite
thf(fact_1095_local_Oprod_Oinfinite,axiom,
! [A2: set_nat,G: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 )
= one2 ) ) ).
% local.prod.infinite
thf(fact_1096_local_Oprod_Odelta_H,axiom,
! [S: set_a,A: a,B: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta'
thf(fact_1097_local_Oprod_Odelta_H,axiom,
! [S: set_b,A: b,B: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta'
thf(fact_1098_local_Oprod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( A = K2 ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta'
thf(fact_1099_local_Oprod_Odelta,axiom,
! [S: set_a,A: a,B: a > a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups2061451144089001601od_a_a @ times @ one2
@ ^ [K2: a] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta
thf(fact_1100_local_Oprod_Odelta,axiom,
! [S: set_b,A: b,B: b > a] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups2061451144089001602od_a_b @ times @ one2
@ ^ [K2: b] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta
thf(fact_1101_local_Oprod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [K2: nat] : ( if_a @ ( K2 = A ) @ ( B @ K2 ) @ one2 )
@ S )
= one2 ) ) ) ) ).
% local.prod.delta
thf(fact_1102_local_Oprod_Oinsert,axiom,
! [A2: set_a,X3: a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X3 @ A2 )
=> ( ( groups2061451144089001601od_a_a @ times @ one2 @ G @ ( insert_a @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001601od_a_a @ times @ one2 @ G @ A2 ) ) ) ) ) ).
% local.prod.insert
thf(fact_1103_local_Oprod_Oinsert,axiom,
! [A2: set_b,X3: b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X3 @ A2 )
=> ( ( groups2061451144089001602od_a_b @ times @ one2 @ G @ ( insert_b @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups2061451144089001602od_a_b @ times @ one2 @ G @ A2 ) ) ) ) ) ).
% local.prod.insert
thf(fact_1104_local_Oprod_Oinsert,axiom,
! [A2: set_nat,X3: nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X3 @ A2 )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( insert_nat @ X3 @ A2 ) )
= ( times @ ( G @ X3 ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ A2 ) ) ) ) ) ).
% local.prod.insert
thf(fact_1105_local_Oprod_OatMost__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% local.prod.atMost_Suc
thf(fact_1106_local_Oprod_Oeq__sum,axiom,
! [I3: set_b,P2: b > a] :
( ( finite_finite_b @ I3 )
=> ( ( groups4667919067926330667od_a_b @ times @ one2 @ P2 @ I3 )
= ( groups2061451144089001602od_a_b @ times @ one2 @ P2 @ I3 ) ) ) ).
% local.prod.eq_sum
thf(fact_1107_local_Oprod_Oeq__sum,axiom,
! [I3: set_nat,P2: nat > a] :
( ( finite_finite_nat @ I3 )
=> ( ( groups7824906719281202852_a_nat @ times @ one2 @ P2 @ I3 )
= ( groups1957776620359388557_a_nat @ times @ one2 @ P2 @ I3 ) ) ) ).
% local.prod.eq_sum
thf(fact_1108_local_Oprod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > a] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one2 ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% local.prod.cl_ivl_Suc
thf(fact_1109_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1110_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1111_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1112_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1113_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1114_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1115_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1116_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1117_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_1118_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1119_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
? [C5: nat] :
( B6
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1120_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1121_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1122_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1123_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1124_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1125_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1126_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1127_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1128_cancel__comm__monoid__add__class_Oadd__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% cancel_comm_monoid_add_class.add_implies_diff
thf(fact_1129_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1130_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1131_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1132_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1133_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1134_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1135_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1136_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1137_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1138_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1139_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1140_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1141_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1142_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1143_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1144_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1145_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1146_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1147_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1148_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1149_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1150_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1151_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1152_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1153_semiring__class_Odistrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% semiring_class.distrib_left
thf(fact_1154_semiring__class_Odistrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% semiring_class.distrib_right
thf(fact_1155_semiring__class_Ocombine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% semiring_class.combine_common_factor
thf(fact_1156_cancel__semigroup__add__class_Oadd__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% cancel_semigroup_add_class.add_right_imp_eq
thf(fact_1157_cancel__semigroup__add__class_Oadd__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% cancel_semigroup_add_class.add_left_imp_eq
thf(fact_1158_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_1159_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B6: nat] : ( plus_plus_nat @ B6 @ A4 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_1160_semigroup__add__class_Oadd_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% semigroup_add_class.add.assoc
thf(fact_1161_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1162_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1163_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1164_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1165_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1166_comm__monoid__add__class_Oadd_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% comm_monoid_add_class.add.comm_neutral
thf(fact_1167_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1168_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1169_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1170_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1171_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1172_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1173_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1174_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1175_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1176_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1177_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1178_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1179_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1180_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1181_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1182_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1183_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1184_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1185_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1186_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1187_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1188_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1189_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1190_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1191_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1192_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1193_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1194_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1195_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1196_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1197_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1198_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1199_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1200_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1201_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1202_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1203_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1204_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1205_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1206_numeral__class_Oone__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% numeral_class.one_plus_numeral_commute
thf(fact_1207_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1208_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1209_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1210_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1211_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1212_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1213_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1214_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1215_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1216_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1217_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1218_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1219_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1220_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1221_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1222_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1223_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1224_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1225_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1226_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1227_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1228_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1229_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1230_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1231_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1232_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1233_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1234_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1235_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1236_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1237_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1238_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1239_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1240_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1241_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1242_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1243_local_Oprod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% local.prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_1244_local_Osum_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P2: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P2 )
=> ( ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ N @ P2 ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ M @ P2 ) ) ) ) ) ).
% local.sum.atLeastLessThan_concat
thf(fact_1245_local_Oprod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P2: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P2 )
=> ( ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ N @ P2 ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ M @ P2 ) ) ) ) ) ).
% local.prod.atLeastLessThan_concat
thf(fact_1246_local_Osum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% local.sum.shift_bounds_Suc_ivl
thf(fact_1247_local_Oprod_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > a,M: nat,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% local.prod.shift_bounds_Suc_ivl
thf(fact_1248_local_Osum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > a,M: nat,K: nat,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% local.sum.shift_bounds_nat_ivl
thf(fact_1249_local_Oprod_Oshift__bounds__nat__ivl,axiom,
! [G: nat > a,M: nat,K: nat,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% local.prod.shift_bounds_nat_ivl
thf(fact_1250_local_Osum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% local.sum.atLeast0_lessThan_Suc
thf(fact_1251_local_Osum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > a,K: nat] :
( ( ( F @ zero_zero_nat )
= zero )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups5773243554134465322_a_nat @ plus @ zero @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% local.sum_shift_lb_Suc0_0_upt
thf(fact_1252_local_Oprod_OatLeast0__lessThan__Suc,axiom,
! [G: nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% local.prod.atLeast0_lessThan_Suc
thf(fact_1253_local_Osum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_nat @ M @ N )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus @ ( G @ M ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% local.sum.atLeast_Suc_lessThan
thf(fact_1254_local_Oprod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_nat @ M @ N )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times @ ( G @ M ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% local.prod.atLeast_Suc_lessThan
thf(fact_1255_local_Osum_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: nat > a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% local.sum.atLeastLessThan_Suc
thf(fact_1256_local_Oprod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: nat > a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% local.prod.atLeastLessThan_Suc
thf(fact_1257_local_Osum_Ohead__if,axiom,
! [N: nat,M: nat,G: nat > a] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% local.sum.head_if
thf(fact_1258_local_Oprod_Ohead__if,axiom,
! [N: nat,M: nat,G: nat > a] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one2 ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% local.prod.head_if
thf(fact_1259_local_Osum_Olast__plus,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus @ ( G @ N ) @ ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% local.sum.last_plus
thf(fact_1260_local_Oprod_Olast__plus,axiom,
! [M: nat,N: nat,G: nat > a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times @ ( G @ N ) @ ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% local.prod.last_plus
thf(fact_1261_local_Osum_OatLeastLessThan__rev,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% local.sum.atLeastLessThan_rev
thf(fact_1262_local_Osum_Onested__swap,axiom,
! [A: nat > nat > a,N: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( groups5773243554134465322_a_nat @ plus @ zero @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [J2: nat] :
( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( A @ I2 @ J2 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% local.sum.nested_swap
thf(fact_1263_local_Oprod_OatLeastLessThan__rev,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2 @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% local.prod.atLeastLessThan_rev
thf(fact_1264_local_Oprod_Onested__swap,axiom,
! [A: nat > nat > a,N: nat] :
( ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( groups1957776620359388557_a_nat @ times @ one2 @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups1957776620359388557_a_nat @ times @ one2
@ ^ [J2: nat] :
( groups1957776620359388557_a_nat @ times @ one2
@ ^ [I2: nat] : ( A @ I2 @ J2 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% local.prod.nested_swap
thf(fact_1265_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1266_finite__atLeastLessThan,axiom,
! [L2: nat,U2: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U2 ) ) ).
% finite_atLeastLessThan
thf(fact_1267_local_Osum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G: nat > a,N: nat,M: nat] :
( ( groups5773243554134465322_a_nat @ plus @ zero @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups5773243554134465322_a_nat @ plus @ zero
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% local.sum.atLeastLessThan_rev_at_least_Suc_atMost
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( power_a @ one2 @ times @ ( groups1779759026887736869um_a_b @ plus @ zero @ f @ a2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] : ( times @ ( f @ I2 ) @ ( f @ I2 ) )
@ a2 )
@ ( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [I2: b] :
( groups1779759026887736869um_a_b @ plus @ zero
@ ^ [J2: b] : ( times @ ( f @ I2 ) @ ( f @ J2 ) )
@ ( minus_minus_set_b @ a2 @ ( insert_b @ I2 @ bot_bot_set_b ) ) )
@ a2 ) ) ) ).
%------------------------------------------------------------------------------