
theorem Th52:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V, a being Real st a <> 1 & a * v in v + 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 <> 1 and
A2: a * v in v + W;
A3: a - 1 <> 0 by A1;
  consider u being VECTOR of V such that
A4: a * v = v + u and
A5: u in W by A2;
  u = u + 0.V by RLVECT_1:4
    .= u + (v - v) by RLVECT_1:15
    .= a * v - v by A4,RLVECT_1:def 3
    .= a * v - 1 * v by RLVECT_1:def 8
    .= (a - 1) * v by RLVECT_1:35;
  then (a - 1)" * u = ((a - 1)" * (a - 1)) * v by RLVECT_1:def 7;
  then 1 * v = (a - 1)" * u by A3,XCMPLX_0:def 7;
  then v = (a - 1)" * u by RLVECT_1:def 8;
  hence thesis by A5,Th15;
end;
