
theorem :: PROPOSITION 4.12.(vi)
  for L be algebraic lower-bounded sup-Semilattice for x be Element of L
  holds compactbelow x is principal Ideal of CompactSublatt L iff x is compact
proof
  let L be algebraic lower-bounded sup-Semilattice;
  let x be Element of L;
  thus compactbelow x is principal Ideal of CompactSublatt L implies x is
  compact
  proof
    assume compactbelow x is principal Ideal of CompactSublatt L;
    then consider y be Element of CompactSublatt L such that
A1: y in compactbelow x and
A2: y is_>=_than compactbelow x by WAYBEL_0:def 21;
    reconsider y9 = y as Element of L by YELLOW_0:58;
    compactbelow x is Subset of CompactSublatt L by Th2;
    then y9 is_>=_than compactbelow x by A2,YELLOW_0:62;
    then y9 >= sup compactbelow x by YELLOW_0:32;
    then
A3: x <= y9 by WAYBEL_8:def 3;
    y9 <= x & y9 is compact by A1,WAYBEL_8:4;
    hence thesis by A3,ORDERS_2:2;
  end;
  thus x is compact implies compactbelow x is principal Ideal of
  CompactSublatt L
  proof
    reconsider I = compactbelow x as non empty Subset of CompactSublatt L by
Th2;
    assume
A4: x is compact;
    then reconsider x9 = x as Element of CompactSublatt L by WAYBEL_8:def 1;
    compactbelow x is non empty directed Subset of L by WAYBEL_8:def 4;
    then reconsider I as non empty directed Subset of CompactSublatt L by
WAYBEL10:23;
    now
      let y,z be Element of CompactSublatt L;
      reconsider y9 = y, z9 = z as Element of L by YELLOW_0:58;
      assume y in I & z <= y;
      then y9 <= x & z9 <= y9 by WAYBEL_8:4,YELLOW_0:59;
      then
A5:   z9 <= x by ORDERS_2:3;
      z9 is compact by WAYBEL_8:def 1;
      hence z in I by A5,WAYBEL_8:4;
    end;
    then reconsider I as Ideal of CompactSublatt L by WAYBEL_0:def 19;
    sup compactbelow x is_>=_than compactbelow x by YELLOW_0:32;
    then x is_>=_than I by WAYBEL_8:def 3;
    then
A6: x9 is_>=_than I by YELLOW_0:61;
    x <= x;
    then x9 in I by A4,WAYBEL_8:4;
    hence thesis by A6,WAYBEL_0:def 21;
  end;
end;
