begin
:: deftheorem Def1 defines are_commutative LOPBAN_4:def 1 :
Lm1:
for X being Banach_Algebra
for z being Element of X
for n being Element of NAT holds
( z * (z #N n) = z #N (n + 1) & (z #N n) * z = z #N (n + 1) & z * (z #N n) = (z #N n) * z )
Lm2:
for X being Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
( w * (z #N n) = (z #N n) * w & z * (w #N n) = (w #N n) * z )
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
:: deftheorem Def2 defines rExpSeq LOPBAN_4:def 2 :
theorem Th13:
:: deftheorem Def3 defines Coef LOPBAN_4:def 3 :
:: deftheorem Def4 defines Coef_e LOPBAN_4:def 4 :
:: deftheorem Def5 defines Shift LOPBAN_4:def 5 :
:: deftheorem Def6 defines Expan LOPBAN_4:def 6 :
:: deftheorem Def7 defines Expan_e LOPBAN_4:def 7 :
:: deftheorem Def8 defines Alfa LOPBAN_4:def 8 :
:: deftheorem Def9 defines Conj LOPBAN_4:def 9 :
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
theorem Th32:
theorem Th33:
Lm3:
for X being Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum (z rExpSeq )) * (Sum (w rExpSeq )) = Sum ((z + w) rExpSeq )
:: deftheorem Def10 defines exp_ LOPBAN_4:def 10 :
:: deftheorem defines exp LOPBAN_4:def 11 :
theorem
theorem Th35:
theorem
theorem Th37:
theorem Th38:
theorem
theorem Th40:
theorem