
theorem
  for L being complete Lattice holds Top L*' = Top L & (Top L)% is
  meet-irreducible
proof
  let L be complete Lattice;
  set X = {d where d is Element of L : Top L [= d & d <> Top L};
A1: X = {}
  proof
    assume X <> {};
    then reconsider X as non empty set;
    set x = the Element of X;
    x in X;
    then consider x9 being Element of L such that
    x9 = x and
A2: Top L [= x9 & x9 <> Top L;
    x9 [= Top L by LATTICES:19;
    hence thesis by A2,LATTICES:8;
  end;
A3: for b being Element of L st b is_less_than {} holds b [= Top L by
LATTICES:19;
  for q being Element of L st q in {} holds Top L [= q;
  then
A4: Top L is_less_than {} by LATTICE3:def 16;
  Top (LattPOSet L) = "/\"({},LattPOSet L) by YELLOW_0:def 12
    .= "/\"({},L) by YELLOW_0:29
    .= Top L by A4,A3,LATTICE3:34;
  then (Top L)% = Top (LattPOSet L) by LATTICE3:def 3;
  hence thesis by A1,A4,A3,LATTICE3:34,WAYBEL_6:10;
end;
