let K be Field; :: thesis: for V being non trivial VectSp of K

for v being Vector of V

for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let V be non trivial VectSp of K; :: thesis: for v being Vector of V

for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let v be Vector of V; :: thesis: for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let a be Scalar of ; :: thesis: for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let W be Linear_Compl of Lin {v}; :: thesis: ( v <> 0. V implies (coeffFunctional (v,W)) . (a * v) = a )

assume A1: v <> 0. V ; :: thesis: (coeffFunctional (v,W)) . (a * v) = a

set cf = coeffFunctional (v,W);

thus (coeffFunctional (v,W)) . (a * v) = a * ((coeffFunctional (v,W)) . v) by HAHNBAN1:def 8

.= a * (1_ K) by A1, Def8

.= a ; :: thesis: verum

for v being Vector of V

for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let V be non trivial VectSp of K; :: thesis: for v being Vector of V

for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let v be Vector of V; :: thesis: for a being Scalar of

for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let a be Scalar of ; :: thesis: for W being Linear_Compl of Lin {v} st v <> 0. V holds

(coeffFunctional (v,W)) . (a * v) = a

let W be Linear_Compl of Lin {v}; :: thesis: ( v <> 0. V implies (coeffFunctional (v,W)) . (a * v) = a )

assume A1: v <> 0. V ; :: thesis: (coeffFunctional (v,W)) . (a * v) = a

set cf = coeffFunctional (v,W);

thus (coeffFunctional (v,W)) . (a * v) = a * ((coeffFunctional (v,W)) . v) by HAHNBAN1:def 8

.= a * (1_ K) by A1, Def8

.= a ; :: thesis: verum