
theorem Th44:
  for V being RealUnitarySpace, W being Subspace of V, v being
VECTOR of V, a being Real st a <> 0 & (a * v) + W = the carrier of W holds v in
  W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  let a be Real;
  assume that
A1: a <> 0 and
A2: (a * v) + W = the carrier of W;
  assume not v in W;
  then not 1 * v in W by RLVECT_1:def 8;
  then not (a" * a) * v in W by A1,XCMPLX_0:def 7;
  then not a" * (a * v) in W by RLVECT_1:def 7;
  then
A3: not a * v in W by Th15;
  0.V in W & a * v + 0.V = a * v by Th11,RLVECT_1:4;
  then a * v in {a * v + u where u is VECTOR of V : u in W};
  hence contradiction by A2,A3;
end;
