let K be non empty doubleLoopStr ; :: thesis: for V, W being non empty ModuleStr over K
for f being Form of V,W
for a being Element of K
for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W
for a being Element of K
for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
let f be Form of V,W; :: thesis: for a being Element of K
for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
let a be Element of K; :: thesis: for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
let w be Vector of W; :: thesis: FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))

now :: thesis: for v being Vector of V holds (FunctionalSAF ((a * f),w)) . v = (a * (FunctionalSAF (f,w))) . v

let v be Vector of V; :: thesis: (FunctionalSAF ((a * f),w)) . v = (a * (FunctionalSAF (f,w))) . v
thus (FunctionalSAF ((a * f),w)) . v = (a * f) . (v,w) by Th9
.= a * (f . (v,w)) by Def3
.= a * ((FunctionalSAF (f,w)) . v) by Th9
.= (a * (FunctionalSAF (f,w))) . v by HAHNBAN1:def 6 ; :: thesis: verum

end;

hence FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w)) by FUNCT_2:63; :: thesis: verum