
theorem Th5:
  for L1 be lower-bounded non empty reflexive antisymmetric RelStr
  for L2 be non empty reflexive antisymmetric RelStr st the RelStr of L1 = the
  RelStr of L2 & L1 is up-complete holds the carrier of CompactSublatt L1 = the
  carrier of CompactSublatt L2
proof
  let L1 be lower-bounded non empty reflexive antisymmetric RelStr;
  let L2 be non empty reflexive antisymmetric RelStr;
  assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is up-complete;
  now
    reconsider L29 = L2 as lower-bounded non empty reflexive antisymmetric
    RelStr by A1,YELLOW_0:13;
    let x be object;
    assume
A3: x in the carrier of CompactSublatt L2;
    then x is Element of CompactSublatt L29;
    then reconsider x2 = x as Element of L2 by YELLOW_0:58;
    reconsider x1 = x2 as Element of L1 by A1;
A4: x2 is compact by A3,WAYBEL_8:def 1;
    L2 is up-complete by A1,A2,WAYBEL_8:15;
    then x1 is compact by A1,A4,WAYBEL_8:9;
    hence x in the carrier of CompactSublatt L1 by WAYBEL_8:def 1;
  end;
  then
A5: the carrier of CompactSublatt L2 c= the carrier of CompactSublatt L1;
  now
    let x be object;
    assume
A6: x in the carrier of CompactSublatt L1;
    then reconsider x1 = x as Element of L1 by YELLOW_0:58;
    reconsider x2 = x1 as Element of L2 by A1;
    x1 is compact by A6,WAYBEL_8:def 1;
    then x2 is compact by A1,A2,WAYBEL_8:9;
    hence x in the carrier of CompactSublatt L2 by WAYBEL_8:def 1;
  end;
  then
  the carrier of CompactSublatt L1 c= the carrier of CompactSublatt L2;
  hence thesis by A5,XBOOLE_0:def 10;
end;
