reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;
reserve F,F1,F2 for filtering Operation of X;
reserve i for Element of NAT;
reserve X for ConstructorDB, A for FinSequence of the Constrs of X,
  x for Element of X;
reserve Y for ref-finite ConstructorDB,
  B for FinSequence of the Constrs of Y,
  y for Element of Y;

theorem Th70:
  n <= m implies ATLEAST-(A,n) c= ATLEAST-(A,m)
  proof
    assume
A1: n <= m;
    let z be object; assume
A2: z in ATLEAST-(A,n); then
    z in {x: card((rng A)\x ref) <= n} by Def34; then
    consider x such that
A3: z = x & card((rng A)\x ref) <= n;
    card((rng A)\x ref) <= m by A1,A3,XXREAL_0:2; then
    x in {x1 where x1 is Element of X: card((rng A)\x1 ref) <= m};
    hence thesis by A2,A3,Def34;
  end;
