
theorem Th42:
  for V be VectSp of F_Complex, v,w be Vector of V for f be
sesquilinear-Form of V,V, r be Real, a be Element of F_Complex st |.a.|
=1 holds f.(v-[**r,0**]*a*w, v-[**r,0**]*a*w) = f.(v,v) - [**r,0**]*(a* f.(w,v)
  ) - [**r,0**]*(a*'*f.(v,w)) + [**r^2,0**]*f.(w,w)
proof
  let V be VectSp of F_Complex, v1,w be Vector of V, f be sesquilinear-Form of
  V,V, r be Real, a be Element of F_Complex such that
A1: |.a.| =1;
  set r1 = [**r,0**]*a;
  set v3 = f.(v1,v1), v4 = f.(v1,w), w1 = f.(w,v1), w2 = f.(w,w);
A2: [**r,0**]*a*([**r,0**]*a*'*w2) = [**r^2,0**]*(a*a*')*w2
    .= [**r^2*1^2,0**]*w2 by A1,Th13;
  f.(v1-r1*w,v1-r1*w) = v3 - r1*'*v4 -(r1*w1 - r1*(r1*'*w2)) by Th38
    .= v3 - (([**r,0**])*'*a*')*v4 - ([**r,0**]*a*w1 - [**r,0**]*a*(([**r,0
  **]*a)*'*w2)) by COMPLFLD:54
    .= v3 - ([**r,0**]*a*')*v4 - ([**r,0**]*a*w1 - [**r,0**]*a*(([**r,0**]*a
  )*'*w2)) by Th12,COMPLEX1:12
    .= v3 - ([**r,0**]*a*')*v4 - ([**r,0**]*a*w1 - [**r,0**]*a*((([**r,0**])
  *'*a*')*w2)) by COMPLFLD:54
    .= v3 - [**r,0**]*(a*'*v4) - ([**r,0**]*a*w1 - [**r,0**]*a*(([**r,0**]*a
  *')*w2)) by Th12,COMPLEX1:12
    .= v3 - [**r,0**]*(a*'*v4) - [**r,0**]*(a*w1)+[**r,0**]*a*(([**r,0**]*a
  *')*w2) by RLVECT_1:29
    .= v3 - [**r,0**]*(a*w1) - [**r,0**]*(a*'*v4)+[**r,0**]*a*(([**r,0**]*a
  *')*w2);
  hence thesis by A2;
end;
