theorem Th44:
  for P st for p, q st p in P & q in P holds dom p = dom q
      holds P is antichain-like
proof
  let P;
  assume that A1: for p, q st p in P & q in P holds dom p = dom q;
  for p, q st p in P & p^q in P holds p = p^q
    proof
    let p, q;
    assume that A2: p in P & p^q in P;
    set r = p^q;
    dom p = dom r by A1, A2;
    then p = r | (dom r) by FINSEQ_1:21 .= r;
    hence thesis;
    end;
  hence thesis by Th40;
end;
