
theorem Th50:
  for V being RealUnitarySpace, W being Subspace of V, u,v1,v2
  being VECTOR of V st u in v1 + W & u in v2 + W holds v1 + W = v2 + W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v1,v2 be VECTOR of V;
  assume that
A1: u in v1 + W and
A2: u in v2 + W;
  consider x1 being VECTOR of V such that
A3: u = v1 + x1 and
A4: x1 in W by A1;
  consider x2 being VECTOR of V such that
A5: u = v2 + x2 and
A6: x2 in W by A2;
  thus v1 + W c= v2 + W
  proof
    let x be object;
    assume x in v1 + W;
    then consider u1 being VECTOR of V such that
A7: x = v1 + u1 and
A8: u1 in W;
    x2 - x1 in W by A4,A6,Th17;
    then
A9: (x2 - x1) + u1 in W by A8,Th14;
    u - x1 = v1 + (x1 - x1) by A3,RLVECT_1:def 3
      .= v1 + 0.V by RLVECT_1:15
      .= v1 by RLVECT_1:4;
    then x = (v2 + (x2 - x1)) + u1 by A5,A7,RLVECT_1:def 3
      .= v2 + ((x2 - x1) + u1) by RLVECT_1:def 3;
    hence thesis by A9;
  end;
  let x be object;
  assume x in v2 + W;
  then consider u1 being VECTOR of V such that
A10: x = v2 + u1 and
A11: u1 in W;
  x1 - x2 in W by A4,A6,Th17;
  then
A12: (x1 - x2) + u1 in W by A11,Th14;
  u - x2 = v2 + (x2 - x2) by A5,RLVECT_1:def 3
    .= v2 + 0.V by RLVECT_1:15
    .= v2 by RLVECT_1:4;
  then x = (v1 + (x1 - x2)) + u1 by A3,A10,RLVECT_1:def 3
    .= v1 + ((x1 - x2) + u1) by RLVECT_1:def 3;
  hence thesis by A12;
end;
