
theorem Th19:
  for T being non empty TopSpace, V being Element of InclPoset the
  topology of T holds V is prime iff for X, Y being Element of InclPoset the
  topology of T st X/\Y c= V holds X c= V or Y c= V
proof
  let T be non empty TopSpace, V be Element of InclPoset the topology of T;
  hereby
    assume
A1: V is prime;
    let X, Y be Element of InclPoset the topology of T;
    assume
A2: X/\Y c= V;
    X/\Y = X"/\"Y by Th18;
    then X"/\"Y <= V by A2,YELLOW_1:3;
    then X <= V or Y <= V by A1;
    hence X c= V or Y c= V by YELLOW_1:3;
  end;
  assume
A3: for X, Y being Element of InclPoset the topology of T st X/\Y c= V
  holds X c= V or Y c= V;
  let X, Y be Element of InclPoset the topology of T such that
A4: X "/\" Y <= V;
  X/\Y = X"/\"Y by Th18;
  then X/\Y c= V by A4,YELLOW_1:3;
  then X c= V or Y c= V by A3;
  hence X <= V or Y <= V by YELLOW_1:3;
end;
