
theorem Th63:
  for V being RealUnitarySpace, W1,W2 being strict Subspace of V,
  u,v being VECTOR of V st v + W1 = u + W2 holds W1 = W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be strict Subspace of V;
  let u,v be VECTOR of V;
  assume
A1: v + W1 = u + W2;
  set V2 = the carrier of W2;
  set V1 = the carrier of W1;
  assume
A2: W1 <> W2;
A3: now
    set x = the Element of V1 \ V2;
    assume V1 \ V2 <> {};
    then x in V1 by XBOOLE_0:def 5;
    then
A4: x in W1;
    then x in V by Th2;
    then reconsider x as Element of V;
    set z = v + x;
    z in u + W2 by A1,A4;
    then consider u1 being VECTOR of V such that
A5: z = u + u1 and
A6: u1 in W2;
    x = 0.V + x by RLVECT_1:4
      .= v - v + x by RLVECT_1:15
      .= - v + (u + u1) by A5,RLVECT_1:def 3;
    then
A7: (v + (- v + (u + u1))) + W1 = v + W1 by A4,Th46;
    v + (- v + (u + u1)) = (v - v) + (u + u1) by RLVECT_1:def 3
      .= 0.V + (u + u1) by RLVECT_1:15
      .= u + u1 by RLVECT_1:4;
    then (u + u1) + W2 = (u + u1) + W1 by A1,A6,A7,Th46;
    hence thesis by A2,Th62;
  end;
A8: now
    set x = the Element of V2 \ V1;
    assume V2 \ V1 <> {};
    then x in V2 by XBOOLE_0:def 5;
    then
A9: x in W2;
    then x in V by Th2;
    then reconsider x as Element of V;
    set z = u + x;
    z in v + W1 by A1,A9;
    then consider u1 being VECTOR of V such that
A10: z = v + u1 and
A11: u1 in W1;
    x = 0.V + x by RLVECT_1:4
      .= u - u + x by RLVECT_1:15
      .= - u + (v + u1) by A10,RLVECT_1:def 3;
    then
A12: (u + (- u + (v + u1))) + W2 = u + W2 by A9,Th46;
    u + (- u + (v + u1)) = (u - u) + (v + u1) by RLVECT_1:def 3
      .= 0.V + (v + u1) by RLVECT_1:15
      .= v + u1 by RLVECT_1:4;
    then (v + u1) + W1 = (v + u1) + W2 by A1,A11,A12,Th46;
    hence thesis by A2,Th62;
  end;
  V1 <> V2 by A2,Th24;
  then not V1 c= V2 or not V2 c= V1;
  hence thesis by A3,A8,XBOOLE_1:37;
end;
