A1: ( [:REAL,X1:] c= [:REAL, the carrier of X:] & dom the Mult of X = [:REAL, the carrier of X:] ) by FUNCT_2:def 1, ZFMISC_1:95;

A2: now :: thesis: for z being object st z in [:REAL,X1:] holds
( the Mult of X | [:REAL,X1:]) . z in X1

let z be object ; :: thesis: ( z in [:REAL,X1:] implies ( the Mult of X | [:REAL,X1:]) . z in X1 )
assume A3: z in [:REAL,X1:] ; :: thesis: ( the Mult of X | [:REAL,X1:]) . z in X1
then consider r, x being object such that
A4: r in REAL and
A5: x in X1 and
A6: z = [r,x] by ZFMISC_1:def 2;
reconsider r = r as Real by A4;
reconsider y = x as VECTOR of X by A5;
[r,x] in dom ( the Mult of X | [:REAL,X1:]) by A1, A3, A6, RELAT_1:62;
then ( the Mult of X | [:REAL,X1:]) . z = r * y by A6, FUNCT_1:47;
hence ( the Mult of X | [:REAL,X1:]) . z in X1 by A5, Def1; :: thesis: verum

end;

dom ( the Mult of X | [:REAL,X1:]) = [:REAL,X1:] by A1, RELAT_1:62;
hence the Mult of X | [:REAL,X1:] is Function of [:REAL,X1:],X1 by A2, FUNCT_2:3; :: thesis: verum