now

let h be convergent_to_0 Complex_Sequence; :: thesis: ( (h " ) (#) ((R1 (#) R2) /* h) is convergent & lim ((h " ) (#) ((R1 (#) R2) /* h)) = 0c )
for seq being Complex_Sequence st seq is non-empty holds
seq " is non-empty

proof

let seq be Complex_Sequence; :: thesis: ( seq is non-empty implies seq " is non-empty )
assume that
A6: seq is non-empty and
A7: not seq " is non-empty ; :: thesis: contradiction
consider n1 being Element of NAT such that
A8: (seq " ) . n1 = 0c by A7, COMSEQ_1:4;
(seq . n1) " = (seq " ) . n1 by VALUED_1:10;
hence contradiction by A6, A8, COMSEQ_1:4, XCMPLX_1:203; :: thesis: verum

end;

then A9: h " is non-empty ;
A10: (h " ) (#) ((R1 (#) R2) /* h) = ((R1 /* h) (#) (R2 /* h)) /" h by CFCONT_1:29
.= (((R1 /* h) (#) (R2 /* h)) (#) (h " )) /" (h (#) (h " )) by A9, COMSEQ_1:40
.= (((R1 /* h) (#) (R2 /* h)) (#) (h " )) (#) (((h " ) " ) (#) (h " )) by COMSEQ_1:33
.= (h (#) (h " )) (#) ((R1 /* h) (#) ((h " ) (#) (R2 /* h))) by COMSEQ_1:11
.= ((h (#) (h " )) (#) (R1 /* h)) (#) ((h " ) (#) (R2 /* h)) by COMSEQ_1:11
.= (h (#) ((h " ) (#) (R1 /* h))) (#) ((h " ) (#) (R2 /* h)) by COMSEQ_1:11 ;
A11: (h " ) (#) (R1 /* h) is convergent by Def3;
then A12: h (#) ((h " ) (#) (R1 /* h)) is convergent by COMSEQ_2:29;
( lim ((h " ) (#) (R1 /* h)) = 0c & lim h = 0c ) by Def1, Def3;
then A13: lim (h (#) ((h " ) (#) (R1 /* h))) = 0 * 0 by A11, COMSEQ_2:30
.= 0c ;
A14: (h " ) (#) (R2 /* h) is convergent by Def3;
hence (h " ) (#) ((R1 (#) R2) /* h) is convergent by A12, A10, COMSEQ_2:29; :: thesis: lim ((h " ) (#) ((R1 (#) R2) /* h)) = 0c
lim ((h " ) (#) (R2 /* h)) = 0c by Def3;
hence lim ((h " ) (#) ((R1 (#) R2) /* h)) = 0c * 0c by A14, A12, A13, A10, COMSEQ_2:30
.= 0c ;
:: thesis: verum

end;

hence for b₁ being PartFunc of COMPLEX ,COMPLEX st b₁ = R1 (#) R2 holds
( b₁ is total & b₁ is REST-like ) by Def3; :: thesis: verum