let K be non empty multMagma ; :: thesis: for V, W being non empty VectSpStr of K
for f being Functional of V
for g being Functional of W
for v being Vector of V holds FunctionalFAF ((FormFunctional (f,g)),v) = (f . v) * g
let V, W be non empty VectSpStr of K; :: thesis: for f being Functional of V
for g being Functional of W
for v being Vector of V holds FunctionalFAF ((FormFunctional (f,g)),v) = (f . v) * g
let f be Functional of V; :: thesis: for g being Functional of W
for v being Vector of V holds FunctionalFAF ((FormFunctional (f,g)),v) = (f . v) * g
let h be Functional of W; :: thesis: for v being Vector of V holds FunctionalFAF ((FormFunctional (f,h)),v) = (f . v) * h
let v be Vector of V; :: thesis: FunctionalFAF ((FormFunctional (f,h)),v) = (f . v) * h
set F = FormFunctional (f,h);
set FF = FunctionalFAF ((FormFunctional (f,h)),v);

now

let y be Vector of W; :: thesis: (FunctionalFAF ((FormFunctional (f,h)),v)) . y = ((f . v) * h) . y
thus (FunctionalFAF ((FormFunctional (f,h)),v)) . y = (FormFunctional (f,h)) . (v,y) by Th9
.= (f . v) * (h . y) by Def11
.= ((f . v) * h) . y by HAHNBAN1:def 6 ; :: thesis: verum

end;

hence FunctionalFAF ((FormFunctional (f,h)),v) = (f . v) * h by FUNCT_2:63; :: thesis: verum