let V, W be non empty ModuleStr over INT.Ring ; :: thesis: for a, b being Element of F_Real
for f being FrForm of V,W holds (a + b) * f = (a * f) + (b * f)
let r, s be Element of F_Real; :: thesis: for f being FrForm of V,W holds (r + s) * f = (r * f) + (s * f)
let f be FrForm of V,W; :: thesis: (r + s) * f = (r * f) + (s * f)

now :: thesis: for v being Vector of V
for w being Vector of W holds ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)
let w be Vector of W; :: thesis: ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)
thus ((r + s) * f) . (v,w) = (r + s) * (f . (v,w)) by Def3
.= (r * (f . (v,w))) + (s * (f . (v,w)))
.= ((r * f) . (v,w)) + (s * (f . (v,w))) by Def3
.= ((r * f) . (v,w)) + ((s * f) . (v,w)) by Def3
.= ((r * f) + (s * f)) . (v,w) by Def2 ; :: thesis: verum

end;

hence (r + s) * f = (r * f) + (s * f) ; :: thesis: verum