reserve a for set;

theorem
  for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L
  holds AR-below <= IdsMap L
proof
  let L be lower-bounded sup-Semilattice, AR be auxiliary Relation of L;
  let x be set;
  assume x in the carrier of L;
  then reconsider x9 = x as Element of L;
A1: (AR-below).x9 = AR-below x9 by Def12;
  (IdsMap L).x9 = downarrow x9 by YELLOW_2:def 4;
  then
A2: (AR-below).x c= (IdsMap L).x by A1,Th12;
  reconsider a = (AR-below).x9,
  b = (IdsMap L).x9 as Element of InclPoset Ids L;
  take a, b;
  thus thesis by A2,YELLOW_1:3;
end;
