reserve X for set;

theorem
  for L be RelStr st for x,y be Element of L holds x <= y iff x c= y
  holds the InternalRel of L = RelIncl the carrier of L
proof
  let L be RelStr;
  assume
A1: for x,y be Element of L holds x <= y iff x c= y;
A2: now
    let Y,Z be set;
    assume Y in the carrier of L & Z in the carrier of L;
    then reconsider Y9 = Y, Z9 = Z as Element of L;
    thus [Y,Z] in the InternalRel of L implies Y c= Z
    proof
      assume [Y,Z] in the InternalRel of L;
      then Y9 <= Z9 by ORDERS_2:def 5;
      hence thesis by A1;
    end;
    thus Y c= Z implies [Y,Z] in the InternalRel of L
    proof
      assume Y c= Z;
      then Y9 <= Z9 by A1;
      hence thesis by ORDERS_2:def 5;
    end;
  end;
  for x be object st x in the carrier of L ex y be object st [x,y] in the
  InternalRel of L
  proof
    let x be object;
    assume x in the carrier of L;
    then reconsider x9 = x as Element of L;
    take y = x9;
    x9 <= y by A1;
    hence thesis by ORDERS_2:def 5;
  end;
  then
A3: dom the InternalRel of L = the carrier of L by RELSET_1:9;
  for y be object st y in the carrier of L
    ex x be object st [x,y] in the InternalRel of L
  proof
    let y be object;
    assume y in the carrier of L;
    then reconsider y9 = y as Element of L;
    take x = y9;
    x <= y9 by A1;
    hence thesis by ORDERS_2:def 5;
  end;
  then field the InternalRel of L = (the carrier of L) \/ the carrier of L by
A3,RELSET_1:10;
  hence thesis by A2,WELLORD2:def 1;
end;
