
theorem Th5:
  for L be non empty reflexive RelStr for x be Element of L holds
  compactbelow x = downarrow x /\ the carrier of CompactSublatt L
proof
  let L be non empty reflexive RelStr;
  let x be Element of L;
  now
    let y be object;
    assume
A1: y in downarrow x /\ the carrier of CompactSublatt L;
    then reconsider y9 = y as Element of L;
    y in downarrow x by A1,XBOOLE_0:def 4;
    then
A2: y9 <= x by WAYBEL_0:17;
    y in the carrier of CompactSublatt L by A1,XBOOLE_0:def 4;
    then y9 is compact by Def1;
    hence y in compactbelow x by A2;
  end;
  then
A3: downarrow x /\ the carrier of CompactSublatt L c= compactbelow x;
  now
    let y be object;
    assume y in compactbelow x;
    then consider y9 be Element of L such that
A4: y9 = y and
A5: y9 <= x & y9 is compact;
    y9 in downarrow x & y9 in the carrier of CompactSublatt L by A5,Def1,
WAYBEL_0:17;
    hence y in downarrow x /\ the carrier of CompactSublatt L by A4,
XBOOLE_0:def 4;
  end;
  then compactbelow x c= downarrow x /\ the carrier of CompactSublatt L;
  hence thesis by A3,XBOOLE_0:def 10;
end;
