theorem Th45:
  for P, a st for p st p in P holds dom p = a holds P is antichain-like
proof
  let P, a;
  assume that A1: for p st p in P holds dom p = a;
  for p, q st p in P & q in P holds dom p = dom q
  proof
    let p, q;
    assume that A2: p in P and A3: q in P;
    dom p = a by A1, A2 .= dom q by A1, A3;
    hence thesis;
  end;
  hence thesis by Th44;
end;
