let z, w be Element of COMPLEX ; :: thesis: 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 ; :: 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_1:31;
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 A4: (k + 1) -' l = (k - l) + 1
.= (k -' l) + 1 by A1, XREAL_1:235 ;
then A5: (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 A4, COMSEQ_3:def 7
.= (((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 A5; :: thesis: verum