consider b1 being Complex such that
A1: for z being Complex holds L /. z = b1 * z by Def4;

now

let h be convergent_to_0 Complex_Sequence; :: thesis: ( (h " ) (#) ((R (#) L) /* h) is convergent & lim ((h " ) (#) ((R (#) L) /* h)) = 0c )
A2: (h " ) (#) ((R (#) L) /* h) = (h " ) (#) ((R /* h) (#) (L /* h)) by CFCONT_1:29
.= ((h " ) (#) (R /* h)) (#) (L /* h) by COMSEQ_1:11 ;
A3: ( h is convergent & lim h = 0c ) by Def1;

now

let n be Element of NAT ; :: thesis: (L /* h) . n = (b1 (#) h) . n
thus (L /* h) . n = L /. (h . n) by FUNCT_2:192
.= b1 * (h . n) by A1
.= (b1 (#) h) . n by VALUED_1:6 ; :: thesis: verum

end;

then A4: L /* h = b1 (#) h by FUNCT_2:113;
then A5: L /* h is convergent ;
A6: lim (L /* h) = b1 * 0c by A3, A4, COMSEQ_2:18
.= 0c ;
A7: ( (h " ) (#) (R /* h) is convergent & lim ((h " ) (#) (R /* h)) = 0c ) by Def3;
hence (h " ) (#) ((R (#) L) /* h) is convergent by A2, A5, COMSEQ_2:29; :: thesis: lim ((h " ) (#) ((R (#) L) /* h)) = 0c
thus lim ((h " ) (#) ((R (#) L) /* h)) = 0c * 0c by A2, A5, A6, A7, COMSEQ_2:30
.= 0c ; :: thesis: verum

end;

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