
theorem
  for T being non empty TopSpace, V being Element of InclPoset the
  topology of T holds V is co-prime iff for X, Y being Element of InclPoset the
  topology of T st V c= X \/ Y holds V c= X or V c= Y
proof
  let T be non empty TopSpace, V be Element of InclPoset the topology of T;
  hereby
    assume
A1: V is co-prime;
    let X, Y be Element of InclPoset the topology of T;
    assume
A2: V c= X \/ Y;
    X \/ Y = X "\/" Y by Th18;
    then V <= X"\/"Y by A2,YELLOW_1:3;
    then V <= X or V <= Y by A1,Th14;
    hence V c= X or V c= Y by YELLOW_1:3;
  end;
  assume
A3: for X, Y being Element of InclPoset the topology of T st V c= X \/ Y
  holds V c= X or V c= Y;
  now
    let X, Y be Element of InclPoset the topology of T such that
A4: V <= X"\/"Y;
    X \/ Y = X"\/"Y by Th18;
    then V c= X \/ Y by A4,YELLOW_1:3;
    then V c= X or V c= Y by A3;
    hence V <= X or V <= Y by YELLOW_1:3;
  end;
  hence thesis by Th14;
end;
