now

let h be convergent_to_0 Complex_Sequence; :: thesis: ( (h " ) (#) ((a (#) R) /* h) is convergent & lim ((h " ) (#) ((a (#) R) /* h)) = 0c )
A1: (h " ) (#) ((a (#) R) /* h) = (h " ) (#) (a (#) (R /* h)) by CFCONT_1:30
.= a (#) ((h " ) (#) (R /* h)) by COMSEQ_1:16 ;
A2: (h " ) (#) (R /* h) is convergent by Def3;
hence (h " ) (#) ((a (#) R) /* h) is convergent by A1; :: thesis: lim ((h " ) (#) ((a (#) R) /* h)) = 0c
lim ((h " ) (#) (R /* h)) = 0c by Def3;
hence lim ((h " ) (#) ((a (#) R) /* h)) = a * 0c by A2, A1, COMSEQ_2:18
.= 0c ;
:: thesis: verum

end;

hence for b₁ being PartFunc of COMPLEX ,COMPLEX st b₁ = a (#) R holds
( b₁ is total & b₁ is REST-like ) by Def3; :: thesis: verum