reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;

theorem
  (.I \/ J.> = { r : ex p,q st r [= p"\/"q & p in I & q in J }
proof
A1: J.: = J;
  (.I \/ J.> = <.(I \/ J).:.) & I.: = I by Th36;
  then
A2: (.I \/ J.> = { r9: ex p9,q9 st p9"/\"q9 [= r9 & p9 in I & q9 in J } by A1,
FILTER_0:35;
  thus (.I \/ J.> c= { r : ex p,q st r [= p"\/"q & p in I & q in J }
  proof
    let x be object;
    assume x in (.I \/ J.>;
    then consider r9 such that
A3: x = r9 and
A4: ex p9,q9 st p9"/\"q9 [= r9 & p9 in I & q9 in J by A2;
    consider p9,q9 such that
A5: p9"/\"q9 [= r9 and
A6: p9 in I & q9 in J by A4;
A7: p9"/\"q9 = .:p9 "\/" .: q9;
    .:r9 [= .:(p9"/\"q9) by A5,LATTICE2:39;
    hence thesis by A3,A6,A7;
  end;
  let x be object;
  assume x in { r : ex p,q st r [= p"\/"q & p in I & q in J };
  then consider r such that
A8: x = r and
A9: ex p,q st r [= p"\/"q & p in I & q in J;
  consider p,q such that
A10: r [= p"\/"q and
A11: p in I & q in J by A9;
A12: p"\/"q = p.:"/\" q.:;
  (p"\/"q).: [= r.: by A10,LATTICE2:38;
  hence thesis by A2,A8,A11,A12;
end;
