let K be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive 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 Th36
.= ((f . v,y) - (f . v,z)) - (f . w,(y - z)) by Th37
.= ((f . v,y) - (f . v,z)) - ((f . w,y) - (f . w,z)) by Th37 ; :: thesis: verum