
theorem
  for L being complete Lattice for a being Element of L st a is
  co-atomic holds a is completely-meet-irreducible
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : a [= x & x <> a};
  assume a is co-atomic;
  then
A1: a is-lower-neighbour-of Top L;
  then
A2: a <> Top L;
A3: for x being object holds x in X implies x in {Top L}
  proof
    let x be object;
    assume x in X;
    then
A4: ex x9 being Element of L st x9 = x & a [= x9 & x9 <> a;
    then reconsider x as Element of L;
    x [= Top L by LATTICES:19;
    then x = Top L by A1,A4;
    hence thesis by TARSKI:def 1;
  end;
A5: a [= Top L by A1;
A6: for x being object holds x in {Top L} implies x in X
  proof
    let x be object;
    assume x in {Top L};
    then x = Top L by TARSKI:def 1;
    hence thesis by A2,A5;
  end;
  Top L = "/\"({Top L},L) by LATTICE3:42
    .= a*' by A3,A6,TARSKI:2;
  hence thesis by A2;
end;
