let K be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr ; :: thesis: for V, W being VectSp of K

for v, u being Vector of V

for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - u),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (u,w)) - (f . (u,t)))

let V, W be VectSp of K; :: thesis: for v, u being Vector of V

for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - u),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (u,w)) - (f . (u,t)))

let v, w be Vector of V; :: thesis: for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - w),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (w,w)) - (f . (w,t)))

let y, z be Vector of W; :: thesis: for f being bilinear-Form of V,W holds f . ((v - w),(y - z)) = ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z)))

let f be bilinear-Form of V,W; :: thesis: f . ((v - w),(y - z)) = ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z)))

set v1 = f . (v,y);

set v3 = f . (v,z);

set v4 = f . (w,y);

set v5 = f . (w,z);

thus f . ((v - w),(y - z)) = (f . (v,(y - z))) - (f . (w,(y - z))) by Th35

.= ((f . (v,y)) - (f . (v,z))) - (f . (w,(y - z))) by Th36

.= ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z))) by Th36 ; :: thesis: verum

for v, u being Vector of V

for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - u),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (u,w)) - (f . (u,t)))

let V, W be VectSp of K; :: thesis: for v, u being Vector of V

for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - u),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (u,w)) - (f . (u,t)))

let v, w be Vector of V; :: thesis: for w, t being Vector of W

for f being bilinear-Form of V,W holds f . ((v - w),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (w,w)) - (f . (w,t)))

let y, z be Vector of W; :: thesis: for f being bilinear-Form of V,W holds f . ((v - w),(y - z)) = ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z)))

let f be bilinear-Form of V,W; :: thesis: f . ((v - w),(y - z)) = ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z)))

set v1 = f . (v,y);

set v3 = f . (v,z);

set v4 = f . (w,y);

set v5 = f . (w,z);

thus f . ((v - w),(y - z)) = (f . (v,(y - z))) - (f . (w,(y - z))) by Th35

.= ((f . (v,y)) - (f . (v,z))) - (f . (w,(y - z))) by Th36

.= ((f . (v,y)) - (f . (v,z))) - ((f . (w,y)) - (f . (w,z))) by Th36 ; :: thesis: verum