let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for v being Vector of V
for w being Vector of W
for a being Element of INT.Ring
for f being FrForm 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 INT.Ring
for f being FrForm 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 INT.Ring
for f being FrForm of V,W st f is homogeneousFAF holds
f . (v,(a * y)) = a * (f . (v,y))
let r be Element of INT.Ring; :: thesis: for f being FrForm of V,W st f is homogeneousFAF holds
f . (v,(r * y)) = r * (f . (v,y))
let f be FrForm of V,W; :: thesis: ( f is homogeneousFAF implies f . (v,(r * y)) = r * (f . (v,y)) )
set F = FrFunctionalFAF (f,v);
assume f is homogeneousFAF ; :: thesis: f . (v,(r * y)) = r * (f . (v,y))
then A1: FrFunctionalFAF (f,v) is homogeneous ;
thus f . (v,(r * y)) = (FrFunctionalFAF (f,v)) . (r * y) by HTh8
.= r * ((FrFunctionalFAF (f,v)) . y) by A1
.= r * (f . (v,y)) by HTh8 ; :: thesis: verum