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

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

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

end;

then reconsider f = f as Linear_Combination of V by Def1;
take f ; :: thesis: for v being Element of V holds f . v = a * (L . v)
let v be Element of V; :: thesis: f . v = a * (L . v)
thus f . v = a * (L . v) by A1; :: thesis: verum