
theorem
  for L being distributive continuous lower-bounded LATTICE st for p
  being Element of L st p is pseudoprime holds p is prime holds L-waybelow is
  multiplicative
proof
  let L be distributive continuous lower-bounded LATTICE such that
A1: for p being Element of L st p is pseudoprime holds p is prime;
  given a,x,y being Element of L such that
A2: [a,x] in L-waybelow & [a,y] in L-waybelow and
A3: not [a,x"/\"y] in L-waybelow;
  now
    let z be object;
    assume that
A4: z in waybelow (x"/\"y) and
A5: z in uparrow a;
    reconsider z as Element of L by A4;
    z << x"/\"y & z >= a by A4,A5,WAYBEL_0:18,WAYBEL_3:7;
    then a << x"/\"y by WAYBEL_3:2;
    hence contradiction by A3,WAYBEL_4:def 1;
  end;
  then waybelow (x"/\"y) misses uparrow a by XBOOLE_0:3;
  then consider P being Ideal of L such that
A6: P is prime and
A7: waybelow (x"/\"y) c= P and
A8: P misses uparrow a by Th23;
  set p = sup P;
  p is pseudoprime by A6;
  then
A9: p is prime by A1;
  a <= a;
  then
A10: a in uparrow a by WAYBEL_0:18;
A11: x"/\"y = sup waybelow (x"/\"y) & ex_sup_of waybelow (x"/\"y), L by
WAYBEL_0:75,WAYBEL_3:def 5;
  ex_sup_of P, L by WAYBEL_0:75;
  then x"/\"y <= p by A7,A11,YELLOW_0:34;
  then x <= p & a << x or y <= p & a << y by A2,A9,WAYBEL_4:def 1;
  then a in P by WAYBEL_3:20;
  hence thesis by A8,A10,XBOOLE_0:3;
end;
