let z, w be Element of COMPLEX ; for l, k being Element of NAT st l <= k holds
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
let l, k be Element of 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_1:31;
then A3:
l <= k + 1
by A1, XXREAL_0:2;
(k + 1) -' l = (k + 1) - l
by A1, A2, XREAL_1:235, XXREAL_0:2;
then A5: (k + 1) -' l =
(k - l) + 1
.=
(k -' l) + 1
by A1, XREAL_1:235
;
then A6: (Alfa ((k + 1),z,w)) . l =
((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . ((k -' l) + 1))
by A3, Def15
.=
((z ExpSeq) . l) * (((Partial_Sums (w ExpSeq)) . (k -' l)) + ((w ExpSeq) . ((k + 1) -' l)))
by A5, 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, Def15
;
((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l)) =
((z |^ l) / (l !c)) * ((w ExpSeq) . ((k + 1) -' l))
by Def8
.=
((z |^ l) / (l !c)) * ((w |^ ((k + 1) -' l)) / (((k + 1) -' l) !c))
by Def8
.=
(((z |^ l) * (w |^ ((k + 1) -' l))) * 1r) / ((l !c) * (((k + 1) -' l) !c))
by XCMPLX_1:77
.=
((z |^ l) * (w |^ ((k + 1) -' l))) * (1r / ((l !c) * (((k + 1) -' l) !c)))
.=
((Coef_e (k + 1)) . l) * ((z |^ l) * (w |^ ((k + 1) -' l)))
by A3, Def11
.=
(((Coef_e (k + 1)) . l) * (z |^ l)) * (w |^ ((k + 1) -' l))
.=
(Expan_e ((k + 1),z,w)) . l
by A3, Def14
;
hence
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
by A6; verum