theorem Th63:
  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;
