reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem ::3.12 (1-5), p.70
  for L being Boolean LATTICE for l being Element of L st l <> Top L
  holds l is prime iff for x be Element of L st x > l holds x = Top L
proof
  let L be Boolean LATTICE, l be Element of L;
  assume
A1: l <> Top L;
  thus l is prime implies for x be Element of L st x > l holds x = Top L
  proof
    assume
A2: l is prime;
    let x be Element of L;
    consider y being Element of L such that
A3: y is_a_complement_of x by WAYBEL_1:def 24;
    x "/\" y = Bottom L by A3;
    then x "/\" y <= l by YELLOW_0:44;
    then
A4: x <= l or y <= l by A2;
    assume x > l;
    then y < x by A4,ORDERS_2:7;
    then
A5: y <= x by ORDERS_2:def 6;
    x "\/" y = Top L by A3;
    hence thesis by A5,YELLOW_0:24;
  end;
  thus (for x be Element of L st x > l holds x = Top L) implies l is prime
  proof
    assume
A6: for z be Element of L st z > l holds z = Top L;
    let x,y be Element of L;
    assume x "/\" y <= l;
    then
A7: l = l "\/" (x "/\" y) by YELLOW_0:24
      .= (l "\/" x) "/\" (l "\/" y) by WAYBEL_1:5;
    assume that
A8: not x <= l and
A9: not y <= l;
A10: l <> (l "\/" y) by A9,YELLOW_0:24;
    l <= (l "\/" y) by A7,YELLOW_0:23;
    then l < (l "\/" y) by A10,ORDERS_2:def 6;
    then
A11: (l "\/" y) = Top L by A6;
A12: l <> (l "\/" x) by A8,YELLOW_0:24;
    l <= (l "\/" x) by A7,YELLOW_0:23;
    then l < (l "\/" x) by A12,ORDERS_2:def 6;
    then (l "\/" x) = Top L by A6;
    hence contradiction by A1,A7,A11,YELLOW_5:2;
  end;
end;
