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 consider v1 such that
A1: u = v + v1 and
A2: v1 in W;
    take x = - v1;
    thus x in W by A2,Th22;
    thus thesis by A1;
  end;
  given v1 such that
A3: v1 in W and
A4: u = v - v1;
  - v1 in W by A3,Th22;
  hence thesis by A4;
end;
