let K be non empty multMagma ; :: thesis: for V, W being non empty ModuleStr over K

for v being Vector of V

for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let V, W be non empty ModuleStr over K; :: thesis: for v being Vector of V

for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let v be Vector of V; :: thesis: for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let y be Vector of W; :: thesis: for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * y)) = a * (f . (v,y))

let r be Element of K; :: thesis: for f being Form of V,W st f is homogeneousFAF holds

f . (v,(r * y)) = r * (f . (v,y))

let f be Form of V,W; :: thesis: ( f is homogeneousFAF implies f . (v,(r * y)) = r * (f . (v,y)) )

set F = FunctionalFAF (f,v);

assume f is homogeneousFAF ; :: thesis: f . (v,(r * y)) = r * (f . (v,y))

then A1: FunctionalFAF (f,v) is homogeneous ;

thus f . (v,(r * y)) = (FunctionalFAF (f,v)) . (r * y) by Th8

.= r * ((FunctionalFAF (f,v)) . y) by A1

.= r * (f . (v,y)) by Th8 ; :: thesis: verum

for v being Vector of V

for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let V, W be non empty ModuleStr over K; :: thesis: for v being Vector of V

for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let v be Vector of V; :: thesis: for w being Vector of W

for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * w)) = a * (f . (v,w))

let y be Vector of W; :: thesis: for a being Element of K

for f being Form of V,W st f is homogeneousFAF holds

f . (v,(a * y)) = a * (f . (v,y))

let r be Element of K; :: thesis: for f being Form of V,W st f is homogeneousFAF holds

f . (v,(r * y)) = r * (f . (v,y))

let f be Form of V,W; :: thesis: ( f is homogeneousFAF implies f . (v,(r * y)) = r * (f . (v,y)) )

set F = FunctionalFAF (f,v);

assume f is homogeneousFAF ; :: thesis: f . (v,(r * y)) = r * (f . (v,y))

then A1: FunctionalFAF (f,v) is homogeneous ;

thus f . (v,(r * y)) = (FunctionalFAF (f,v)) . (r * y) by Th8

.= r * ((FunctionalFAF (f,v)) . y) by A1

.= r * (f . (v,y)) by Th8 ; :: thesis: verum