theorem Th43: :: HERMITAN:43

for V being VectSp of F_Complex
for v, w being Vector of V
for f being diagReR+0valued hermitan-Form of V
for r being Real
for a being Element of F_Complex st |.a.| = 1 & Re (a * (f . (w,v))) = |.(f . (w,v)).| holds
( Re (f . ((v - (([**r,0**] * a) * w)),(v - (([**r,0**] * a) * w)))) = ((signnorm (f,v)) - ((2 * |.(f . (w,v)).|) * r)) + ((signnorm (f,w)) * (r ^2)) & 0 <= ((signnorm (f,v)) - ((2 * |.(f . (w,v)).|) * r)) + ((signnorm (f,w)) * (r ^2)) )