reconsider a = a as Element of COMPLEX by XCMPLX_0:def 2;
deffunc H₁( Element of V) -> Element of COMPLEX = In ((a * (L . $1)),COMPLEX);
consider f being Function of the carrier of V,COMPLEX such that
A1: for v being Element of V holds f . v = H₁(v) from FUNCT_2:sch 4();
reconsider f = f as Element of Funcs ( the carrier of V,COMPLEX) by FUNCT_2:8;

now :: thesis: for v being VECTOR of V st not v in Carrier L holds
f . v = 0c

let v be VECTOR of V; :: thesis: ( not v in Carrier L implies f . v = 0c )
assume not v in Carrier L ; :: thesis: f . v = 0c
then A2: L . v = 0c ;
thus f . v = H₁(v) by A1
.= 0c by A2 ; :: thesis: verum

end;

then reconsider f = f as C_Linear_Combination of V by Def1;
take f ; :: thesis: for v being VECTOR of V holds f . v = a * (L . v)
let v be VECTOR of V; :: thesis: f . v = a * (L . v)
f . v = H₁(v) by A1;
hence f . v = a * (L . v) ; :: thesis: verum