
theorem Th15: :: PROPOSITION 4.12.(iv)
  for L be lower-bounded sup-Semilattice holds CompactSublatt L is
  lower-bounded sup-Semilattice
proof
  let L be lower-bounded sup-Semilattice;
  ex x being Element of CompactSublatt L st x is_<=_than the carrier of
  CompactSublatt L
  proof
    reconsider x = Bottom L as Element of CompactSublatt L by WAYBEL_8:3;
    take x;
    now
      let a be Element of CompactSublatt L;
A1:   the carrier of CompactSublatt L c= the carrier of L by YELLOW_0:def 13;
      assume a in the carrier of CompactSublatt L;
      reconsider a9 = a as Element of L by A1;
      Bottom L <= a9 by YELLOW_0:44;
      hence x <= a by YELLOW_0:60;
    end;
    hence thesis by LATTICE3:def 8;
  end;
  hence thesis by YELLOW_0:def 4;
end;
