
theorem Th49:

:: Parallerogram equality
  for V be VectSp of F_Complex for f be hermitan-Form of V for v,w
  be Element of V holds signnorm(f,v+w) + signnorm(f,v-w) =2* signnorm(f,v)+ 2*
  signnorm(f,w)
proof
  let V be VectSp of F_Complex, f be hermitan-Form of V, v1,w be Element of V;
  set v3 = f.(v1,v1), v4 = f.(v1,w), w1 = f.(w,v1), w2 = f.(w,w), vp = f.(v1+w
  ,v1+w), vm = f.(v1-w,v1-w), s1 = signnorm(f,v1), s2 = signnorm(f,w), sp =
  signnorm(f,v1+w), sm = signnorm(f,v1-w);
  vp = v3 + v4 +(w1+w2) by BILINEAR:28;
  then
A1: vp + vm = v3 + v4 +(w1+w2) + (v3 - v4 - (w1 -w2)) by Th36
    .= v3 + (v4 + v3 - v4) + (w1+w2) - (w1 -w2)
    .= v3 + v3 + (w1+w2) - (w1 -w2) by COMPLFLD:12
    .= v3 + v3 + (w1+w2 - (w1 -w2))
    .= v3 + v3 + (w1+w2 - w1 +w2) by RLVECT_1:29
    .= v3 + v3 + (w2 + w2) by COMPLFLD:12;
  thus sp + sm = Re (vp+ vm) by HAHNBAN1:10
    .= Re (v3 + v3) + Re (w2 + w2) by A1,HAHNBAN1:10
    .= Re v3 + Re v3 + Re (w2 + w2) by HAHNBAN1:10
    .= 2 * (Re v3) + (Re w2 + Re w2) by HAHNBAN1:10
    .= 2 * s1 + 2 *s2;
end;
