A2: ( [:COMPLEX,V1:] c= [:COMPLEX, the carrier of V:] & dom the Mult of V = [:COMPLEX, the carrier of V:] ) by FUNCT_2:def 1, ZFMISC_1:95;
A3: for z being object st z in [:COMPLEX,V1:] holds
( the Mult of V | [:COMPLEX,V1:]) . z in V1

proof

let z be object ; :: thesis: ( z in [:COMPLEX,V1:] implies ( the Mult of V | [:COMPLEX,V1:]) . z in V1 )
assume A4: z in [:COMPLEX,V1:] ; :: thesis: ( the Mult of V | [:COMPLEX,V1:]) . z in V1
consider r, x being object such that
A5: r in COMPLEX and
A6: x in V1 and
A7: z = [r,x] by A4, ZFMISC_1:def 2;
reconsider r = r as Complex by A5;
reconsider y = x as VECTOR of V by A6;
[r,x] in dom ( the Mult of V | [:COMPLEX,V1:]) by A2, A4, A7, RELAT_1:62;
then ( the Mult of V | [:COMPLEX,V1:]) . z = r * y by A7, FUNCT_1:47;
hence ( the Mult of V | [:COMPLEX,V1:]) . z in V1 by A1, A6; :: thesis: verum

end;

dom ( the Mult of V | [:COMPLEX,V1:]) = [:COMPLEX,V1:] by A2, RELAT_1:62;
hence the Mult of V | [:COMPLEX,V1:] is Function of [:COMPLEX,V1:],V1 by A3, FUNCT_2:3; :: thesis: verum