
theorem Th62:
  for V being RealUnitarySpace, W1,W2 being strict Subspace of V,
  v being VECTOR of V holds v + W1 = v + W2 iff W1 = W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be strict Subspace of V;
  let v be VECTOR of V;
  thus v + W1 = v + W2 implies W1 = W2
  proof
    assume
A1: v + W1 = v + W2;
    the carrier of W1 = the carrier of W2
    proof
A2:   the carrier of W1 c= the carrier of V by Def1;
      thus the carrier of W1 c= the carrier of W2
      proof
        let x be object;
        assume
A3:     x in the carrier of W1;
        then reconsider y = x as Element of V by A2;
        set z = v + y;
        x in W1 by A3;
        then z in v + W2 by A1;
        then consider u being VECTOR of V such that
A4:     z = v + u and
A5:     u in W2;
        y = u by A4,RLVECT_1:8;
        hence thesis by A5;
      end;
      let x be object;
      assume
A6:   x in the carrier of W2;
      the carrier of W2 c= the carrier of V by Def1;
      then reconsider y = x as Element of V by A6;
      set z = v + y;
      x in W2 by A6;
      then z in v + W1 by A1;
      then consider u being VECTOR of V such that
A7:   z = v + u and
A8:   u in W1;
      y = u by A7,RLVECT_1:8;
      hence thesis by A8;
    end;
    hence thesis by Th24;
  end;
  thus thesis;
end;
