reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th22:
  for n be natural number holds
  NAT\{t where t is Element of NAT:n <= t} is finite
  proof
    let n be natural number;
    NAT\{t where t is Element of NAT:n <= t} c= n+1
    proof
      NAT\{t where t is Element of NAT:n <= t} c= Seg n \/{0}
      proof
        set S= NAT\{t where t is Element of NAT:n <= t};
        per cases;
        suppose
A1:       n=0;
          S is empty
          proof
            let x be object such that
A2:         x in S;
            x in NAT & not x in {t where t is Element of NAT:0<=t}
            by A1,A2,XBOOLE_0:def 5;
            hence thesis;
          end;
          hence thesis;
        end;
        suppose
          n>0;
          let x be object such that
A3:       x in S;
A4:       x in NAT & not x in {t where t is Element of NAT:n<=t}
          by A3,XBOOLE_0:def 5;
          reconsider x as Element of NAT by A3;
A5:       x=0 or x in Seg x by FINSEQ_1:3;
          x <= n by A4;
          then Seg x c= Seg n by FINSEQ_1:5;
          then x in {0} or x in Seg n by A5,TARSKI:def 1;
          hence thesis by XBOOLE_0:def 3;
        end;
      end;
      hence thesis by AFINSQ_1:4;
    end;
    hence thesis;
  end;
