deffunc H₁( Element of V) -> Element of the carrier of R = (L . $1) * a;
consider f being Function of V,R 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, the carrier of R) by FUNCT_2:8;

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

let v be Vector of V; :: thesis: ( not v in Carrier L implies f . v = 0. R )
assume not v in Carrier L ; :: thesis: f . v = 0. R
then L . v = 0. R ;
hence f . v = (0. R) * a by A1
.= 0. R ;
:: thesis: verum

end;

then reconsider f = f as Linear_Combination of V by Def2;
take f ; :: thesis: for v being Vector of V holds f . v = (L . v) * a
let v be Vector of V; :: thesis: f . v = (L . v) * a
thus f . v = (L . v) * a by A1; :: thesis: verum