
theorem
  for R being non empty RelStr st the InternalRel of R is_reflexive_in
the carrier of R & the InternalRel of R is antisymmetric transitive holds R is
  reflexive antisymmetric transitive
proof
  let r be non empty RelStr;
  set i = the InternalRel of r;
  set c = the carrier of r;
  assume that
A1: i is_reflexive_in c and
A2: i is antisymmetric transitive;
A3: i is_transitive_in field i by A2;
A4: field i = c by A1,PARTIT_2:21;
  then i is_antisymmetric_in c by A2;
  hence thesis by A1,A4,A3,ORDERS_2:def 2,def 3,def 4;
end;
