
theorem
  for L being complete Lattice for a being Element of L st a is atomic
  holds a is completely-join-irreducible
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : x [= a & x <> a};
  assume a is atomic;
  then
A1: a is-upper-neighbour-of Bottom L;
  then
A2: a <> Bottom L;
A3: for x being object holds x in X implies x in {Bottom L}
  proof
    let x be object;
    assume x in X;
    then
A4: ex x9 being Element of L st x9 = x & x9 [= a & x9 <> a;
    then reconsider x as Element of L;
    Bottom L [= x by LATTICES:16;
    then x = Bottom L by A1,A4;
    hence thesis by TARSKI:def 1;
  end;
A5: Bottom L [= a by A1;
A6: for x being object holds x in {Bottom L} implies x in X
  proof
    let x be object;
    assume x in {Bottom L};
    then x = Bottom L by TARSKI:def 1;
    hence thesis by A2,A5;
  end;
  Bottom L = "\/"({Bottom L},L) by LATTICE3:42
    .= *'a by A3,A6,TARSKI:2;
  hence thesis by A2;
end;
