
theorem Th20:
  for L be Boolean LATTICE for a be Element of L holds a is atom
  iff a is co-prime & a <> Bottom L
proof
  let L be Boolean LATTICE;
  let a be Element of L;
  thus a is atom implies a is co-prime & a <> Bottom L
  proof
    assume
A1: a is atom;
    now
      let x,y be Element of L;
      assume that
A2:   a <= x "\/" y and
A3:   not a <= x;
A4:   a "/\" x <= a by YELLOW_0:23;
      a "/\" x <> a by A3,YELLOW_0:25;
      then not Bottom L < a "/\" x by A1,A4;
      then
A5:   (not Bottom L <= a "/\" x) or not Bottom L <> a "/\" x by ORDERS_2:def 6;
      a = a "/\" (x "\/" y) by A2,YELLOW_0:25
        .= (a "/\" x) "\/" (a "/\" y) by WAYBEL_1:def 3
        .= a "/\" y by A5,WAYBEL_1:3,YELLOW_0:44;
      hence a <= y by YELLOW_0:25;
    end;
    hence a is co-prime by WAYBEL14:14;
    thus thesis by A1;
  end;
  assume that
A6: a is co-prime and
A7: a <> Bottom L;
A8: now
    let b be Element of L;
    assume that
A9: Bottom L < b and
A10: b <= a;
    consider c be Element of L such that
A11: c is_a_complement_of b by WAYBEL_1:def 24;
A12: b "/\" c = Bottom L by A11;
A13: not a <= c
    by A10,ORDERS_2:3,A9,A12,YELLOW_0:25;
    b "\/" c = Top L by A11;
    then a <= b "\/" c by YELLOW_0:45;
    then a <= b by A6,A13,WAYBEL14:14;
    hence b = a by A10,ORDERS_2:2;
  end;
  Bottom L <= a by YELLOW_0:44;
  then Bottom L < a by A7,ORDERS_2:def 6;
  hence thesis by A8;
end;
