let K be non empty left_unital doubleLoopStr ; :: thesis: for V, W being non empty ModuleStr over K
for f being Form of V,W holds (1. K) * f = f
let V, W be non empty ModuleStr over K; :: thesis: for f being Form of V,W holds (1. K) * f = f
let f be Form of V,W; :: thesis: (1. K) * f = f

now :: thesis: for v being Vector of V
for w being Vector of W holds ((1. K) * f) . (v,w) = f . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds ((1. K) * f) . (v,w) = f . (v,w)
let w be Vector of W; :: thesis: ((1. K) * f) . (v,w) = f . (v,w)
thus ((1. K) * f) . (v,w) = (1. K) * (f . (v,w)) by Def3
.= f . (v,w) ; :: thesis: verum

end;

hence (1. K) * f = f ; :: thesis: verum