
theorem
  for L being distributive Lattice,
      a, b being Element of L st not a [= b holds
    ex P being Ideal of L st P is prime & not a in P & b in P
  proof
    let L be distributive Lattice,
        a, b be Element of L;
    assume
AA: not a [= b;
ZZ: <.a.) misses (.b.>
    proof
      assume <.a.) meets (.b.>; then
      consider y being object such that
A2:   y in (.b.> & y in <.a.) by XBOOLE_0:3;
      reconsider y as Element of L by A2;
A4:   y [= b by A2,FILTER_2:28;
      a [= y by A2,FILTER_0:15;
      hence thesis by AA,A4,LATTICES:7;
    end;
    set I1 = (.b.>, F1 = <.a.);
    consider P being Ideal of L such that
B1: P is prime & I1 c= P & P misses F1 by Th15,ZZ;
B2: b in P by B1,FILTER_2:28;
    a in F1; then
    not a in P by B1,XBOOLE_0:3;
    hence thesis by B1,B2;
  end;
