let K be non empty left-distributive doubleLoopStr ; for V, W being non empty ModuleStr over 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 ModuleStr over K; 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; for f being Form of V,W holds (r + s) * f = (r * f) + (s * f)
let f be Form of V,W; (r + s) * f = (r * f) + (s * f)
now 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;
for w being Vector of W holds ((r + s) * f) . (v,w) = ((r * f) + (s * f)) . (v,w)let w be
Vector of
W;
((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)))
by VECTSP_1:def 3
.=
((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
;
verum end;
hence
(r + s) * f = (r * f) + (s * f)
; verum