reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th14:
  for V being ManySortedSet of NAT st
  V.0 = the set of all [{}, i] where i is Element of NAT &
  for n being Nat holds
  V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
  A is finite}
  for i,j being Element of NAT st i <= j holds V.i c= V.j
proof
  let V be ManySortedSet of NAT such that
A1: V.0 = the set of all [{}, i] where i is Element of NAT and
A2: for n being Nat holds
  V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT:
  A is finite};
  defpred Q[Nat] means V.0 c= V.$1;
A3: now
    let j;
    assume Q[j];
A4: V.(j+1) = {[varcl A, k] where A is Subset of V.j, k is Element of NAT:
    A is finite} by A2;
    thus Q[j+1]
    proof
      let x be object;
      assume x in V.0;
      then
A5:   ex i being Element of NAT st x = [{}, i] by A1;
      {} c= V.j;
      hence thesis by A4,A5,Th8;
    end;
  end;
  defpred P[Nat] means for i st i <= $1 holds V.i c= V.$1;
A6: P[ 0 ] by NAT_1:3;
A7: now
    let j be Nat;
    assume
A8: P[j];
A9: V.j c= V.(j+1) proof per cases by NAT_1:6;
      suppose j = 0;
        hence thesis by A3;
      end;
      suppose ex k being Nat st j = k+1;
        then consider k being Nat such that
A10:    j = k+1;
        reconsider k as Element of NAT by ORDINAL1:def 12;
A11:    V.j = {[varcl A, n] where A is Subset of V.k, n is Element of NAT:
        A is finite} by A2,A10;
        A12:    V
.(j+1) = {[varcl A, n] where A is Subset of V.j,n is Element of NAT:
        A is finite} by A2;
A13:    V.k c= V.j by A8,A10,NAT_1:11;
        let x be object;
        assume x in V.j;
        then consider A being Subset of V.k, n being Element of NAT such that
A14:    x = [varcl A, n] and
A15:    A is finite by A11;
        A c= V.j by A13;
        hence thesis by A12,A14,A15;
      end;
    end;
    thus P[j+1]
    proof
      let i;
      assume i <= j+1;
      then i = j+1 or V.i c= V.j by A8,NAT_1:8;
      hence thesis by A9;
    end;
  end;
  for j being Nat holds P[j] from NAT_1:sch 2(A6,A7);
  hence thesis;
end;
