let w, z be Complex; 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; ( 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
; (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;
(k + 1) -' l = (k + 1) - l
by A1, A2, XREAL_1:233, XXREAL_0:2;
then A4: (k + 1) -' l =
(k - l) + 1
.=
(k -' l) + 1
by A1, XREAL_1:233
;
then A5: (Alfa ((k + 1),z,w)) . l =
((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . ((k -' l) + 1))
by A3, Def11
.=
((z ExpSeq) . l) * (((Partial_Sums (w ExpSeq)) . (k -' l)) + ((w ExpSeq) . ((k + 1) -' l)))
by A4, SERIES_1:def 1
.=
(((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . (k -' l))) + (((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l)))
.=
((Alfa (k,z,w)) . l) + (((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l)))
by A1, Def11
;
((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l)) =
((z |^ l) / (l !)) * ((w ExpSeq) . ((k + 1) -' l))
by Def4
.=
((z |^ l) / (l !)) * ((w |^ ((k + 1) -' l)) / (((k + 1) -' l) !))
by Def4
.=
(((z |^ l) * (w |^ ((k + 1) -' l))) * 1r) / ((l !) * (((k + 1) -' l) !))
by XCMPLX_1:76
.=
((z |^ l) * (w |^ ((k + 1) -' l))) * (1r / ((l !) * (((k + 1) -' l) !)))
.=
((Coef_e (k + 1)) . l) * ((z |^ l) * (w |^ ((k + 1) -' l)))
by A3, Def7
.=
(((Coef_e (k + 1)) . l) * (z |^ l)) * (w |^ ((k + 1) -' l))
.=
(Expan_e ((k + 1),z,w)) . l
by A3, Def10
;
hence
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
by A5; verum