theorem
  u in v + W iff ex v1 st v1 in W & u = v + v1
  proof
    thus u in v + W implies ex v1 st v1 in W & u = v + v1
    proof
      assume u in v + W;
      then ex v1 st u = v + v1 & v1 in W;
      hence thesis;
    end;
    given v1 such that
A1: v1 in W & u = v + v1;
    thus thesis by A1;
  end;
