
theorem Th53:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V, a being Real st v in W holds a * v in v + W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  let a be Real;
  assume v in W;
  then
A1: (a - 1) * v in W by Th15;
  a * v = ((a - 1) + 1) * v .= (a - 1) * v + 1 * v by RLVECT_1:def 6
    .= v + (a - 1) * v by RLVECT_1:def 8;
  hence thesis by A1;
end;
