
theorem Th43:
  for V be VectSp of F_Complex, v,w be Vector of V for f be
diagReR+0valued hermitan-Form of V, r be Real, a be 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
proof
  let V be VectSp of F_Complex, v1,w be Vector of V, f be diagReR+0valued
  hermitan-Form of V, r be Real, a be Element of F_Complex such that
A1: |.a.| =1 and
A2: Re (a * f.(w,v1)) = |.f.(w,v1).|;
  set v3=f.(v1,v1), v4=f.(v1,w), w1=f.(w,v1), w2=f.(w,w), A = signnorm(f,v1),
  B = |.w1.|, C = signnorm(f,w);
A3: Re [**r,0**] =r & Im [**r,0**] = 0 by COMPLEX1:12;
  then
A4: Re ([**r,0**]*(a* w1)) = r * B by A2,Th10;
  a*'*v4 = (a*'*v4)*'*' .= (a*'*' * v4*')*' by COMPLFLD:54
    .= (a*w1)*' by Def5;
  then
A5: Re ([**r,0**]*(a*'*v4)) = r * Re ((a*w1)*') by A3,Th10
    .= r * B by A2,COMPLEX1:27;
  Re [**r^2,0**] =r^2 & Im [**r^2,0**] = 0 by COMPLEX1:12;
  then
A6: Re ([**r^2,0**]*w2) = r^2 * C by Th10;
  f.(v1-[**r,0**]*a*w,v1-[**r,0**]*a*w) = v3 - [**r,0**]*(a* w1) - [**r,0
  **]*(a*'*v4) + [**r^2,0**]*w2 by A1,Th42;
  then Re (f.(v1-[**r,0**]*a*w,v1-[**r,0**]*a*w)) = Re(v3 - [**r,0**]*(a* w1)
  - [**r,0**]*(a*'*v4)) + C * r^2 by A6,HAHNBAN1:10
    .= Re(v3 - [**r,0**]*(a* w1)) - r*B + C * r^2 by A5,Th9
    .= A - r*B - r*B + C * r^2 by A4,Th9
    .= A -2*B*r + C * r^2;
  hence thesis by Def7;
end;
