
theorem Th44:
  for L being continuous lower-bounded LATTICE st L-waybelow is
  multiplicative for p being Element of L holds p is pseudoprime iff for a,b
  being Element of L st a"/\"b << p holds a <= p or b <= p
proof
  let L be continuous lower-bounded LATTICE such that
A1: L-waybelow is multiplicative;
  let p be Element of L;
  hereby
    assume
A2: p is pseudoprime;
    let a,b be Element of L;
    assume a"/\"b << p;
    then inf {a,b} << p by YELLOW_0:40;
    then ex c being Element of L st c in {a,b} & c <= p by A2,Th35;
    hence a <= p or b <= p by TARSKI:def 2;
  end;
  assume for a,b being Element of L st a"/\" b << p holds a <= p or b <= p;
  hence thesis by A1,Lm3,Th34;
end;
