A2: [:REAL ,X1:] c= [:REAL ,the carrier of X:] by ZFMISC_1:118;
A3: dom the Mult of X = [:REAL ,the carrier of X:] by FUNCT_2:def 1;
then A4: dom (the Mult of X | [:REAL ,X1:]) = [:REAL ,X1:] by A2, RELAT_1:91;
for z being set st z in [:REAL ,X1:] holds
(the Mult of X | [:REAL ,X1:]) . z in X1

proof

let z be set ; :: thesis: ( z in [:REAL ,X1:] implies (the Mult of X | [:REAL ,X1:]) . z in X1 )
assume A5: z in [:REAL ,X1:] ; :: thesis: (the Mult of X | [:REAL ,X1:]) . z in X1
consider r, x being set such that
A6: ( r in REAL & x in X1 & z = [r,x] ) by A5, ZFMISC_1:def 2;
reconsider y = x as VECTOR of X by A6;
reconsider r = r as Real by A6;
[r,x] in dom (the Mult of X | [:REAL ,X1:]) by A2, A3, A5, A6, RELAT_1:91;
then (the Mult of X | [:REAL ,X1:]) . z = r * y by A6, FUNCT_1:70;
hence (the Mult of X | [:REAL ,X1:]) . z in X1 by A1, A6, RLSUB_1:def 1; :: thesis: verum

end;

hence the Mult of X | [:REAL ,X1:] is Function of [:REAL ,X1:],X1 by A4, FUNCT_2:5; :: thesis: verum