let K be non empty doubleLoopStr ; :: thesis: for V, W being non empty VectSpStr of 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 VectSpStr of 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

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 Th10
.= a * (f . v,w) by Def4
.= a * ((FunctionalSAF f,w) . v) by Th10
.= (a * (FunctionalSAF f,w)) . v by HAHNBAN1:def 9 ; :: thesis: verum

end;

hence FunctionalSAF (a * f),w = a * (FunctionalSAF f,w) by FUNCT_2:113; :: thesis: verum