let V, W be Z_Module; :: thesis: for v, u being Vector of V
for w, t being Vector of W
for a, b being Element of INT.Ring
for f being bilinear-FrForm of V,W holds f . ((v - (a * u)),(w - (b * t))) = ((f . (v,w)) - (b * (f . (v,t)))) - ((a * (f . (u,w))) - (a * (b * (f . (u,t)))))
let v, w be Vector of V; :: thesis: for w, t being Vector of W
for a, b being Element of INT.Ring
for f being bilinear-FrForm of V,W holds f . ((v - (a * w)),(w - (b * t))) = ((f . (v,w)) - (b * (f . (v,t)))) - ((a * (f . (w,w))) - (a * (b * (f . (w,t)))))
let y, z be Vector of W; :: thesis: for a, b being Element of INT.Ring
for f being bilinear-FrForm of V,W holds f . ((v - (a * w)),(y - (b * z))) = ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (a * (b * (f . (w,z)))))
let a, b be Element of INT.Ring; :: thesis: for f being bilinear-FrForm of V,W holds f . ((v - (a * w)),(y - (b * z))) = ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (a * (b * (f . (w,z)))))
let f be bilinear-FrForm of V,W; :: thesis: f . ((v - (a * w)),(y - (b * z))) = ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (a * (b * (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 - (a * w)),(y - (b * z))) = ((f . (v,y)) - (f . (v,(b * z)))) - ((f . ((a * w),y)) - (f . ((a * w),(b * z)))) by HTh37
.= ((f . (v,y)) - (b * (f . (v,z)))) - ((f . ((a * w),y)) - (f . ((a * w),(b * z)))) by HTh32
.= ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (f . ((a * w),(b * z)))) by HTh31
.= ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (a * (f . (w,(b * z))))) by HTh31
.= ((f . (v,y)) - (b * (f . (v,z)))) - ((a * (f . (w,y))) - (a * (b * (f . (w,z))))) by HTh32 ; :: thesis: verum