
theorem Th41:
  for V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over F_Complex for f be diagReR+0valued
diagRvalued Form of V,V for v be Vector of V holds |. f.(v,v) .| = Re (f.(v,v))
  & signnorm (f,v) = |. f.(v,v) .|
proof
  let V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
empty ModuleStr over F_Complex, f be diagReR+0valued diagRvalued Form of V,V,
  v be Vector of V;
  set v1 = f.(v,v);
  0 <= Re v1 & Im v1 = 0 by Def6,Def7;
  hence thesis by Th14;
end;
