let X be Banach_Algebra; :: thesis: for w, z being Element of X
for k, l being Nat st l <= k holds
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
let w, z be Element of X; :: thesis: for k, l being Nat st l <= k holds
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
let k, l be Nat; :: thesis: ( l <= k implies (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) )
assume A1: l <= k ; :: thesis: (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
A2: k < k + 1 by XREAL_1:29;
then A3: l <= k + 1 by A1, XXREAL_0:2;
A4: ((z rExpSeq) . l) * ((w rExpSeq) . ((k + 1) -' l)) = ((1 / (l !)) * (z #N l)) * ((w rExpSeq) . ((k + 1) -' l)) by Def2
.= ((1 / (l !)) * (z #N l)) * ((1 / (((k + 1) -' l) !)) * (w #N ((k + 1) -' l))) by Def2
.= ((1 / (l !)) * (1 / (((k + 1) -' l) !))) * ((z #N l) * (w #N ((k + 1) -' l))) by LOPBAN_3:38
.= (1 / ((l !) * (((k + 1) -' l) !))) * ((z #N l) * (w #N ((k + 1) -' l))) by XCMPLX_1:102
.= ((Coef_e (k + 1)) . l) * ((z #N l) * (w #N ((k + 1) -' l))) by A3, Def4
.= (((Coef_e (k + 1)) . l) * (z #N l)) * (w #N ((k + 1) -' l)) by LOPBAN_3:38
.= (Expan_e ((k + 1),z,w)) . l by A3, Def7 ;
(k + 1) -' l = (k + 1) - l by A1, A2, XREAL_1:233, XXREAL_0:2;
then A5: (k + 1) -' l = (k - l) + 1
.= (k -' l) + 1 by A1, XREAL_1:233 ;
then (Alfa ((k + 1),z,w)) . l = ((z rExpSeq) . l) * ((Partial_Sums (w rExpSeq)) . ((k -' l) + 1)) by A3, Def8
.= ((z rExpSeq) . l) * (((Partial_Sums (w rExpSeq)) . (k -' l)) + ((w rExpSeq) . ((k + 1) -' l))) by A5, BHSP_4:def 1
.= (((z rExpSeq) . l) * ((Partial_Sums (w rExpSeq)) . (k -' l))) + (((z rExpSeq) . l) * ((w rExpSeq) . ((k + 1) -' l))) by LOPBAN_3:38
.= ((Alfa (k,z,w)) . l) + (((z rExpSeq) . l) * ((w rExpSeq) . ((k + 1) -' l))) by A1, Def8 ;
hence (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) by A4; :: thesis: verum