
theorem
  for L being complete Lattice for D being Subset of L st D is
  infimum-dense holds MIRRS(L) c= D
proof
  let L be complete Lattice;
  let D be Subset of L;
  assume
A1: D is infimum-dense;
    let x be object;
    assume x in MIRRS(L);
    then consider x9 being Element of L such that
A2: x9 = x and
A3: x9 is completely-meet-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 & x [= d};
    set X = {d where d is Element of L: x [= d & 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 & x [= x9;
      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 & x [= u9;
      hence thesis by A5,A6;
    end;
    then D9 c= X;
    then "/\"(X,L) [= "/\"(D9,L) by LATTICE3:45;
    then
A7: "/\"(X,L) [= x by A1,Th24;
    for q being Element of L st q in X holds x [= q
    proof
      let q be Element of L;
      assume q in X;
      then ex q9 being Element of L st q9 = q & x [= q9 & q9 <> x;
      hence thesis;
    end;
    then x is_less_than X by LATTICE3:def 16;
    then
A8: x [= "/\"(X,L) by LATTICE3:39;
    (x9)*' <> x9 by A3;
    hence thesis by A2,A7,A8,LATTICES:8;
end;
