reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th37:
  for L being lower-bounded meet-continuous Semilattice, I being Ideal of L
  holds DownMap I is approximating
proof
  let L be lower-bounded meet-continuous Semilattice;
  let I be Ideal of L;
  for x be Element of L ex ii be Subset of L st ii = (DownMap I).x & x = sup ii
  proof
    let x be Element of L;
    set ii = (DownMap I).x;
    ii in the carrier of InclPoset Ids L;
    then ii in Ids L by YELLOW_1:1;
    then consider ii9 be Ideal of L such that
A1: ii9 = ii;
    reconsider dI = ( downarrow x ) /\ I as Subset of L;
    per cases;
    suppose
A2:   x <= sup I;
      then sup ii9 = sup dI by A1,Def16
        .= sup ({x} "/\" I) by Th36
        .= x "/\" sup I by WAYBEL_2:def 6
        .= x by A2,YELLOW_0:25;
      hence thesis by A1;
    end;
    suppose not x <= sup I;
      then sup ii9 = sup downarrow x by A1,Def16
        .= x by WAYBEL_0:34;
      hence thesis by A1;
    end;
  end;
  hence thesis;
end;
