reserve x, y for set;

theorem Th8:
  for P being reflexive non empty RelStr holds P is upper-bounded
  iff the InternalRel of P is upper-bounded
proof
  let P be reflexive non empty RelStr;
  (the carrier of P) \/ the carrier of P = the carrier of P;
  then
A1: field the InternalRel of P c= the carrier of P by RELSET_1:8;
  thus P is upper-bounded implies the InternalRel of P is upper-bounded
  proof
    given x being Element of P such that
A2: x is_>=_than the carrier of P;
    take x;
    let y;
    assume y in field the InternalRel of P;
    then reconsider y as Element of P by A1;
    x >= y by A2;
    hence thesis;
  end;
  set y = the Element of P;
  given x such that
A3: for y st y in field the InternalRel of P holds [y,x] in the
  InternalRel of P;
  y <= y;
  then [y,y] in the InternalRel of P;
  then y in field the InternalRel of P by RELAT_1:15;
  then [y,x] in the InternalRel of P by A3;
  then x in field the InternalRel of P by RELAT_1:15;
  then reconsider x as Element of P by A1;
  take x;
  let y be Element of P;
  y <= y;
  then [y,y] in the InternalRel of P;
  then y in field the InternalRel of P by RELAT_1:15;
  then [y,x] in the InternalRel of P by A3;
  hence thesis;
end;
