
theorem
  for V being RealUnitarySpace, W being Subspace of V, v1,v2 being
VECTOR of V holds (ex C being Coset of W st v1 in C & v2 in C) iff v1 - v2 in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v1,v2 be VECTOR of V;
  thus (ex C being Coset of W st v1 in C & v2 in C) implies v1 - v2 in W
  proof
    given C be Coset of W such that
A1: v1 in C & v2 in C;
    ex v being VECTOR of V st C = v + W by Def5;
    hence thesis by A1,Th59;
  end;
  assume v1 - v2 in W;
  then consider v being VECTOR of V such that
A2: v1 in v + W & v2 in v + W by Th59;
  reconsider C = v + W as Coset of W by Def5;
  take C;
  thus thesis by A2;
end;
