
theorem Th43:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V, a being Real st v in W holds (a * v) + W = the carrier of W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  let a be Real;
  assume
A1: v in W;
  thus (a * v) + W c= the carrier of W
  proof
    let x be object;
    assume x in (a * v) + W;
    then consider u being VECTOR of V such that
A2: x = a * v + u and
A3: u in W;
    a * v in W by A1,Th15;
    then a * v + u in W by A3,Th14;
    hence thesis by A2;
  end;
  let x be object;
  assume
A4: x in the carrier of W;
  then
A5: x in W;
  the carrier of W c= the carrier of V by Def1;
  then reconsider y = x as Element of V by A4;
A6: a * v + (y - a * v) = (y + a * v) - a * v by RLVECT_1:def 3
    .= y + (a * v - a * v) by RLVECT_1:def 3
    .= y + 0.V by RLVECT_1:15
    .= x by RLVECT_1:4;
  a * v in W by A1,Th15;
  then y - a * v in W by A5,Th17;
  hence thesis by A6;
end;
