
theorem
  for L being complete Lattice for D being Subset of L st D is
  supremum-dense holds JIRRS(L) c= D
proof
  let L be complete Lattice;
  let D be Subset of L;
  assume
A1: D is supremum-dense;
  for x being object holds x in JIRRS(L) implies x in D
  proof
    let x be object;
    assume x in JIRRS(L);
    then consider x9 being Element of L such that
A2: x9 = x and
A3: x9 is completely-join-irreducible;
    assume
A4: not x in D;
    reconsider x as Element of L by A2;
    set D9 = {d where d is Element of L: d in D & d [= x};
    set X = {d where d is Element of L: d [= x & d <> x};
A5: not x in D9
    proof
      assume x in D9;
      then ex x9 being Element of L st x9 = x & x9 in D & x9 [= x;
      hence thesis by A4;
    end;
    for u being object holds u in D9 implies u in X
    proof
      let u be object;
      assume
A6:   u in D9;
      then
      ex u9 being Element of L st u9 = u & u9 in D & u9 [= x;
      hence thesis by A5,A6;
    end;
    then D9 c= X;
    then "\/"(D9,L) [= "\/"(X,L) by LATTICE3:45;
    then
A7: x [= "\/"(X,L) by A1,Th23;
    for q being Element of L st q in X holds q [= x
    proof
      let q be Element of L;
      assume q in X;
      then ex q9 being Element of L st q9 = q & q9 [= x & q9 <> x;
      hence thesis;
    end;
    then X is_less_than x by LATTICE3:def 17;
    then
A8: "\/"(X,L) [= x by LATTICE3:def 21;
    *'(x9) <> x9 by A3;
    hence thesis by A2,A7,A8,LATTICES:8;
  end;
  hence thesis;
end;
