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

theorem Th17:
  for A being finite Subset of Vars, i being Nat holds [varcl A, i] in Vars
proof
  let A be finite Subset of Vars, i be Nat;
  consider V being ManySortedSet of NAT such that
A1: Vars = Union V and
A2: V.0 = the set of all [{}, k] where k is Element of NAT and
A3: for n being Nat holds
  V.(n+1) = {[varcl b, j] where b is Subset of V.n, j is Element of NAT:
  b is finite} by Def2;
  consider j being Element of NAT such that
A4: A c= V.j by A2,A3,Th15;
A5: V.(j+1) = {[varcl B, k] where B is Subset of V.j, k is Element of NAT: B
  is finite} by A3;
  i in NAT by ORDINAL1:def 12;
  then
A6: [varcl A, i] in V.(j+1) by A4,A5;
  dom V = NAT by PARTFUN1:def 2;
  hence thesis by A1,A6,CARD_5:2;
end;
