
theorem Th21:
  for A being Universal_Algebra, B being Subset of A
  for n,m being Nat st n <= m holds B|^n c= B|^m
proof
  let A be Universal_Algebra;
  let B be Subset of A;
  let n,m be Nat;
  assume n <= m;
  then
A1: ex i being Nat st ( m = n+i) by NAT_1:10;
  defpred P[Nat] means B|^n c= B|^(n+$1);
A2: P[0];
A3: now
    let i be Nat;
    assume
A4: P[i];
    deffunc Rec(set,set) = $2 \/
    {Den(o,A).p where o is (Element of dom the charact of A),
    p is Element of (the carrier of A)*: p in dom Den(o,A) & rng p c= $2};
    B|^(n+i+1) = Rec(n,B|^(n+i)) by Th19;
    then B|^(n+i) c= B|^(n+(i+1)) by XBOOLE_1:7;
    hence P[i+1] by A4,XBOOLE_1:1;
  end;
  for i being Nat holds P[i] from NAT_1:sch 2(A2,A3);
  hence thesis by A1;
end;
