
theorem Cor16:
  for L being distributive Lattice,
      I being Ideal of L,
      a being Element of L st not a in I
   ex P being Ideal of L st P is prime & I c= P & not a in P
  proof
    let L be distributive Lattice,
        I be Ideal of L,
        a be Element of L;
    assume
A0: not a in I;
    set F = <.a.);
A2: a in F;
    I misses F
    proof
      assume I meets F; then
      consider x being object such that
A1:   x in I & x in F by XBOOLE_0:3;
      reconsider x as Element of L by A1;
      a [= x by A1,FILTER_0:15;
      hence thesis by A0,FILTER_2:21,A1;
    end; then
    consider P being Ideal of L such that
T1: P is prime & I c= P & P misses F by Th15;
    not a in P by A2,T1,XBOOLE_0:3;
    hence thesis by T1;
  end;
