A2: dom the Mult of X = [:COMPLEX, the carrier of X:] by FUNCT_2:def 1;
A3: [:COMPLEX,X1:] c= [:COMPLEX, the carrier of X:] by ZFMISC_1:95;
A4: for z being object st z in [:COMPLEX,X1:] holds
( the Mult of X | [:COMPLEX,X1:]) . z in X1

proof

let z be object ; :: thesis: ( z in [:COMPLEX,X1:] implies ( the Mult of X | [:COMPLEX,X1:]) . z in X1 )
assume A5: z in [:COMPLEX,X1:] ; :: thesis: ( the Mult of X | [:COMPLEX,X1:]) . z in X1
consider r, x being object such that
A6: r in COMPLEX and
A7: x in X1 and
A8: z = [r,x] by A5, ZFMISC_1:def 2;
reconsider r = r as Complex by A6;
reconsider y = x as VECTOR of X by A7;
[r,x] in dom ( the Mult of X | [:COMPLEX,X1:]) by A3, A2, A5, A8, RELAT_1:62;
then ( the Mult of X | [:COMPLEX,X1:]) . z = r * y by A8, FUNCT_1:47;
hence ( the Mult of X | [:COMPLEX,X1:]) . z in X1 by A1, A7; :: thesis: verum

end;

dom ( the Mult of X | [:COMPLEX,X1:]) = [:COMPLEX,X1:] by A3, A2, RELAT_1:62;
hence the Mult of X | [:COMPLEX,X1:] is Function of [:COMPLEX,X1:],X1 by A4, FUNCT_2:3; :: thesis: verum