
theorem

:: Parallerogram equality - second version (stronger assumption)
  for V be VectSp of F_Complex for f be diagReR+0valued hermitan-Form of
V for v,w be Element of V holds |. f.(v+w,v+w) .| + |. f.(v-w,v-w) .| = 2*|. f.
  (v,v) .| + 2*|. f.(w,w) .|
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V, v1,w
  be Element of V;
  set s1 = signnorm(f,v1), s2 = signnorm(f,w), sp = signnorm(f,v1+w), sm =
  signnorm(f,v1-w);
A1: sm = |. f.(v1-w,v1-w) .| & s1 = |. f.(v1,v1) .| by Th41;
  sp + sm = 2*s1 + 2*s2 & sp = |. f.(v1+w,v1+w) .| by Th41,Th49;
  hence thesis by A1,Th41;
end;
