
theorem Th1:
  for L be non empty reflexive transitive RelStr for x,y be Element
  of L holds x <= y implies compactbelow x c= compactbelow y
proof
  let L be non empty reflexive transitive RelStr;
  let x,y be Element of L;
  assume
A1: x <= y;
  now
    let z be object;
    assume z in compactbelow x;
    then z in {v where v is Element of L: x >= v & v is compact} by
WAYBEL_8:def 2;
    then consider z9 be Element of L such that
A2: z9 = z and
A3: x >= z9 and
A4: z9 is compact;
    z9 <= y by A1,A3,ORDERS_2:3;
    hence z in compactbelow y by A2,A4,WAYBEL_8:4;
  end;
  hence thesis;
end;
