
theorem Th72:
  for V being RealUnitarySpace, W being Subspace of V, C being
  Coset of W, u being VECTOR of V holds u in C iff C = u + W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let C be Coset of W;
  let u be VECTOR of V;
  thus u in C implies C = u + W
  proof
    assume
A1: u in C;
    ex v being VECTOR of V st C = v + W by Def5;
    hence thesis by A1,Th48;
  end;
  thus thesis by Th37;
end;
