begin
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem
theorem
:: deftheorem Def1 defines with_the_same_dom MESFUNC8:def 1 :
for X, Y being set
for F being Functional_Sequence of X,Y holds
( F is with_the_same_dom iff rng F is with_common_domain );
:: deftheorem Def2 defines with_the_same_dom MESFUNC8:def 2 :
for X, Y being set
for F being Functional_Sequence of X,Y holds
( F is with_the_same_dom iff for n, m being natural number holds dom (F . n) = dom (F . m) );
:: deftheorem Def3 defines inf MESFUNC8:def 3 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being PartFunc of X,ExtREAL holds
( b3 = inf f iff ( dom b3 = dom (f . 0) & ( for x being Element of X st x in dom b3 holds
b3 . x = inf (f # x) ) ) );
:: deftheorem Def4 defines sup MESFUNC8:def 4 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being PartFunc of X,ExtREAL holds
( b3 = sup f iff ( dom b3 = dom (f . 0) & ( for x being Element of X st x in dom b3 holds
b3 . x = sup (f # x) ) ) );
:: deftheorem Def5 defines inferior_realsequence MESFUNC8:def 5 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being with_the_same_dom Functional_Sequence of X,ExtREAL holds
( b3 = inferior_realsequence f iff for n being natural number holds
( dom (b3 . n) = dom (f . 0) & ( for x being Element of X st x in dom (b3 . n) holds
(b3 . n) . x = (inferior_realsequence (f # x)) . n ) ) );
:: deftheorem Def6 defines superior_realsequence MESFUNC8:def 6 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being with_the_same_dom Functional_Sequence of X,ExtREAL holds
( b3 = superior_realsequence f iff for n being natural number holds
( dom (b3 . n) = dom (f . 0) & ( for x being Element of X st x in dom (b3 . n) holds
(b3 . n) . x = (superior_realsequence (f # x)) . n ) ) );
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
:: deftheorem MESFUNC8:def 7 :
canceled;
:: deftheorem Def8 defines lim_inf MESFUNC8:def 8 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being PartFunc of X,ExtREAL holds
( b3 = lim_inf f iff ( dom b3 = dom (f . 0) & ( for x being Element of X st x in dom b3 holds
b3 . x = lim_inf (f # x) ) ) );
:: deftheorem Def9 defines lim_sup MESFUNC8:def 9 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being PartFunc of X,ExtREAL holds
( b3 = lim_sup f iff ( dom b3 = dom (f . 0) & ( for x being Element of X st x in dom b3 holds
b3 . x = lim_sup (f # x) ) ) );
theorem Th11:
theorem Th12:
theorem
:: deftheorem Def10 defines lim MESFUNC8:def 10 :
for X being non empty set
for f being Functional_Sequence of X,ExtREAL
for b3 being PartFunc of X,ExtREAL holds
( b3 = lim f iff ( dom b3 = dom (f . 0) & ( for x being Element of X st x in dom b3 holds
b3 . x = lim (f # x) ) ) );
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem
theorem Th25:
theorem Th26:
theorem Th27:
begin
theorem Th28:
theorem
theorem