
theorem
  for X be RealUnitarySpace, M be Subspace of X,
      x be Point of X,
      y1,y2,z1,z2 be Point of X st
      y1 in M & y2 in M & z1 in Ort_Comp M & z2 in Ort_Comp M &
      x = y1 + z1 & x = y2 + z2
     holds y1 = y2 & z1 = z2
proof
  let X be RealUnitarySpace, M be Subspace of X,
      x be Point of X;
    let y1,y2,z1,z2 be Point of X;
    assume that
A1: y1 in M & y2 in M & z1 in Ort_Comp M & z2 in Ort_Comp M and
A2: x = y1 + z1 & x = y2 + z2;
    y1 + (z1 + -y2) = (y2 + z2) + -y2 by RLVECT_1:def 3,A2; then
    y1 + (-y2 + z1) = y2 + (-y2 + z2) by RLVECT_1:def 3; then
    (y1 + -y2) + z1 = y2 + (-y2 + z2) by RLVECT_1:def 3; then
    (y1 - y2) + z1 = (y2 + -y2) + z2 by RLVECT_1:def 3; then
    (y1 - y2) + z1 = z2 + 0.X by RLVECT_1:def 10; then
    (y1 - y2) + (z1 + -z1) = z2 + -z1 by RLVECT_1:def 3; then
A31: (y1 - y2) + 0.X = z2 + -z1 by RLVECT_1:def 10;
A4: y1 - y2 in M & z2 - z1 in Ort_Comp M by A1,RUSUB_1:17; then
    y1 - y2 in (the carrier of M) /\ the carrier of (Ort_Comp M)
      by XBOOLE_0:def 4,A31; then
    y1 - y2 = 0.X by Lm814;
    hence y1 = y2 by RLVECT_1:21;
    z2 - z1 in (the carrier of M) /\ the carrier of (Ort_Comp M)
      by A4,A31,XBOOLE_0:def 4; then
    z2 - z1 = 0.X by Lm814;
    hence z1 = z2 by RLVECT_1:21;
end;
