let K be non empty left-distributive doubleLoopStr ; :: thesis: for V, W being non empty VectSpStr of K
for a, b being Element of K
for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)
let V, W be non empty VectSpStr of K; :: thesis: for a, b being Element of K
for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)
let r, s be Element of K; :: thesis: for f being Form of V,W holds (r + s) * f = (r * f) + (s * f)
let f be Form of V,W; :: thesis: (r + s) * f = (r * f) + (s * f)

now

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 Def4
.= (r * (f . v,w)) + (s * (f . v,w)) by VECTSP_1:def 12
.= ((r * f) . v,w) + (s * (f . v,w)) by Def4
.= ((r * f) . v,w) + ((s * f) . v,w) by Def4
.= ((r * f) + (s * f)) . v,w by Def3 ; :: thesis: verum

end;

hence (r + s) * f = (r * f) + (s * f) by BINOP_1:2; :: thesis: verum