theorem
  for V being strict ComplexLinearSpace, W being strict Subspace of V
  holds the carrier of W = the carrier of V implies W = V
proof
  let V be strict ComplexLinearSpace, W be strict Subspace of V;
  assume
A1: the carrier of W = the carrier of V;
  V is Subspace of V by Th44;
  hence thesis by A1,Th49;
end;
