
theorem Th24:
  for L being distributive LATTICE, I being Ideal of L, x being
Element of L st not x in I ex P being Ideal of L st P is prime & I c= P & not x
  in P
proof
  let L be distributive LATTICE, I be Ideal of L, x be Element of L such that
A1: not x in I;
  now
    let a be object;
    assume that
A2: a in I and
A3: a in uparrow x;
    reconsider a as Element of L by A2;
    a >= x by A3,WAYBEL_0:18;
    hence contradiction by A1,A2,WAYBEL_0:def 19;
  end;
  then I misses uparrow x by XBOOLE_0:3;
  then consider P being Ideal of L such that
A4: P is prime & I c= P and
A5: P misses uparrow x by Th23;
  take P;
  thus P is prime & I c= P by A4;
  assume x in P;
  then not x in uparrow x by A5,XBOOLE_0:3;
  then not x <= x by WAYBEL_0:18;
  hence thesis;
end;
