theorem Th42:
  x in v + W iff ex u st u in W & x = v + u
proof
  thus x in v + W implies ex u st u in W & x = v + u
  proof
    assume x in v + W;
    then consider u such that
A1: x = v + u & u in W;
    take u;
    thus thesis by A1;
  end;
  given u such that
A2: u in W & x = v + u;
  thus thesis by A2;
end;
