theorem Th68:
  v in W implies (z * v) + W = the carrier of W
proof
  assume
A1: v in W;
  thus (z * v) + W c= the carrier of W
  proof
    let x be object;
    assume x in (z * v) + W;
    then consider u such that
A2: x = z * v + u and
A3: u in W;
    z * v in W by A1,Th40;
    then z * v + u in W by A3,Th39;
    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 Def8;
  then reconsider y = x as Element of V by A4;
A6: z * v + (y - z * v) = (y + z * v) - z * v by RLVECT_1:def 3
    .= y + (z * v - z * v) by RLVECT_1:def 3
    .= y + 0.V by RLVECT_1:15
    .= x by RLVECT_1:4;
  z * v in W by A1,Th40;
  then y - z * v in W by A5,Th42;
  hence thesis by A6;
end;
