reserve i,j for Nat;

theorem Th7:
 for x being Element of Vars st vars x is natural holds vars x = 0
  proof let x be Element of Vars;
   assume x`1 is natural; then
   reconsider n = x`1 as Element of NAT;
    Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT:
    A is finite} & x in Vars by ABCMIZ_1:18; then
   consider A being Subset of Vars, j being Element of NAT such that
A1: x = [varcl A, j] & A is finite;
   set i = the Element of n;
   assume A2: x`1 <> 0; then
A3: i in n;
   reconsider i as Element of NAT by A2,ORDINAL1:10;
    n = varcl A & vars x c= Vars by A1; then
    i in Vars by A3;
   hence thesis;
  end;
