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 a <= F.
  j) implies a <= Inf F
proof
  let F be Function of J, the carrier of L;
  assume
A1: for j holds a <= F.j;
  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 a <= c by A1;
  end;
  then a is_<=_than rng F;
  hence thesis by YELLOW_0:33;
end;
