
theorem
  for L be Boolean LATTICE holds ATOM L = (PRIME (L opp)) \ {Bottom L}
proof
  let L be Boolean LATTICE;
A1: (PRIME (L opp)) \ {Bottom L} c= ATOM L
  proof
    let x be object;
    assume
A2: x in (PRIME (L opp)) \ {Bottom L};
    then reconsider x9 = x as Element of (L opp);
    x in PRIME (L opp) by A2,XBOOLE_0:def 5;
    then
A3: x9 is prime by WAYBEL_6:def 7;
    not x in {Bottom L} by A2,XBOOLE_0:def 5;
    then x9 <> Bottom L by TARSKI:def 1;
    then
A4: ~x9 <> Bottom L by LATTICE3:def 7;
    (~x9)~ = ~x9 by LATTICE3:def 6
      .= x9 by LATTICE3:def 7;
    then ~x9 is co-prime by A3,WAYBEL_6:def 8;
    then ~x9 is atom by A4,Th20;
    then ~x9 in ATOM L by Def2;
    hence thesis by LATTICE3:def 7;
  end;
  ATOM L c= (PRIME (L opp)) \ {Bottom L}
  proof
    let x be object;
    assume
A5: x in ATOM L;
    then reconsider x9 = x as Element of L;
A6: x9 is atom by A5,Def2;
    then x9~ is prime by WAYBEL_6:def 8;
    then x9~ in PRIME (L opp) by WAYBEL_6:def 7;
    then
A7: x in PRIME (L opp) by LATTICE3:def 6;
    x <> Bottom L by A6;
    then not x in {Bottom L} by TARSKI:def 1;
    hence thesis by A7,XBOOLE_0:def 5;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
