
theorem
  for L being distributive Lattice,
      I being Ideal of L holds
    I = meet { P where P is Ideal of L : P is prime & I c= P }
  proof
    let L be distributive Lattice,
        I be Ideal of L;
X1: [#]L is prime
    proof
      set II = [#]L;
      for p, q being Element of L holds
        p "/\" q in II iff p in II or q in II;
      hence thesis by FILTER_2:def 10;
    end;
    set P = [#]L;
    set X = { P where P is Ideal of L : P is prime & I c= P };
    set I1 = meet X;
k1: P in X by X1;
    set x = the Element of X;
    meet X c= the carrier of L by k1,SETFAM_1:def 1; then
    reconsider I2 = I1 as Subset of L;
    assume I1 <> I; then
    per cases;
    suppose not I1 c= I; then
      consider a being object such that
A1:   a in I1 & not a in I;
      a in I2 by A1; then
      reconsider a as Element of L;
      consider P being Ideal of L such that
A2:   P is prime & I c= P & not a in P by Cor16,A1;
      P in X by A2; then
      I1 c= P by SETFAM_1:3;
      hence thesis by A1,A2;
    end;
    suppose not I c= I1; then
      consider a being object such that
A1:   a in I & not a in I1;
      consider Y being set such that
X4:   Y in X & not a in Y by SETFAM_1:def 1,A1,k1;
      consider P1 being Ideal of L such that
X5:   Y = P1 & P1 is prime & I c= P1 by X4;
      thus thesis by A1,X4,X5;
    end;
  end;
