
theorem
  for L being complete Lattice holds *'Bottom L = Bottom L & (Bottom L)%
  is join-irreducible
proof
  let L be complete Lattice;
  set X = {d where d is Element of L : d [= Bottom L & d <> Bottom 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: x9 [= Bottom L & x9 <> Bottom L;
    Bottom L [= x9 by LATTICES:16;
    hence thesis by A2,LATTICES:8;
  end;
A3: for b being Element of L st b is_greater_than {} holds Bottom L [= b by
LATTICES:16;
A4: for x,y being Element of LattPOSet L st Bottom (LattPOSet L) = x "\/" y
  holds x = Bottom (LattPOSet L) or y = Bottom (LattPOSet L)
  proof
    reconsider L9 = LattPOSet L as lower-bounded antisymmetric non empty
    RelStr;
    let x,y be Element of LattPOSet L;
    reconsider x9 = x as Element of L9;
    reconsider y9 = y as Element of L9;
    assume Bottom (LattPOSet L) = x "\/" y;
    then
A5: Bottom (LattPOSet L) >= x & Bottom (LattPOSet L) >= y by YELLOW_0:22;
    x9 >= Bottom (L9) or y9 >= Bottom (L9) by YELLOW_0:44;
    hence thesis by A5,ORDERS_2:2;
  end;
  for q being Element of L st q in {} holds q [= Bottom L;
  then
A6: Bottom L is_greater_than {} by LATTICE3:def 17;
  Bottom (LattPOSet L) = "\/"({},LattPOSet L) by YELLOW_0:def 11
    .= "\/"({},L) by YELLOW_0:29
    .= Bottom L by A6,A3,LATTICE3:def 21;
  then (Bottom L)% = Bottom (LattPOSet L) by LATTICE3:def 3;
  hence thesis by A1,A6,A3,A4,LATTICE3:def 21,WAYBEL_6:def 3;
end;
