
theorem Th9:
  for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is up-complete for x be Element of L1 for
  y be Element of L2 st x = y & x is compact holds y is compact
proof
  let L1,L2 be non empty reflexive antisymmetric RelStr;
  assume
A1: the RelStr of L1 = the RelStr of L2 & L1 is up-complete;
  let x be Element of L1;
  let y be Element of L2;
  assume that
A2: x = y and
A3: x is compact;
  x << x by A3,WAYBEL_3:def 2;
  then y << y by A1,A2,Th8;
  hence thesis by WAYBEL_3:def 2;
end;
