reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;
reserve J for non empty set,
  j for Element of J;

theorem
  for F being Function of J, the carrier of L holds (for j holds F.j <=
  a) implies Sup F <= a
proof
  let F be Function of J, the carrier of L;
  assume
A1: for j holds F.j <= a;
  now
    let c be Element of L;
    assume c in rng F;
    then consider j being object such that
A2: j in dom F and
A3: c = F.j by FUNCT_1:def 3;
    reconsider j as Element of J by A2;
    c = F.j by A3;
    hence c <= a by A1;
  end;
  then rng F is_<=_than a;
  hence thesis by YELLOW_0:32;
end;
