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

theorem
  for R1, R2 being auxiliary Relation of L st R1 c= R2 holds
  R1-below <= R2-below
proof
  let R1, R2 be auxiliary Relation of L;
  assume
A1: R1 c= R2;
  let x be set;
  assume x in the carrier of L;
  then reconsider x9 = x as Element of L;
A2: R1-below x9 c= R2-below x9
  proof
    let a be object;
    assume a in R1-below x9;
    then ex b be Element of L st ( b = a)&( [b,x9] in R1);
    hence thesis by A1;
  end;
  reconsider A1 = (R1-below).x9, A2 = (R2-below).x9
  as Element of InclPoset Ids L;
  take A1, A2;
A3: (R1-below).x = R1-below x9 by Def12;
  (R2-below).x = R2-below x9 by Def12;
  hence thesis by A2,A3,YELLOW_1:3;
end;
