
theorem Th1:
  for X being finite non empty Subset of NAT holds ex n being
  Element of NAT st X c= Seg n \/ {0}
proof
  let X be finite non empty Subset of NAT;
  reconsider m = max X as Element of NAT by ORDINAL1:def 12;
  take m;
  let x be object;
A1: Seg m c= Seg m \/ {0} by XBOOLE_1:7;
A2: {0} c= Seg m \/ {0} by XBOOLE_1:7;
  assume
A3: x in X;
  then reconsider n = x as Element of NAT;
A4: n <= m by A3,XXREAL_2:def 8;
  per cases by NAT_1:25;
  suppose
    1 <= n;
    then n in Seg m by A4,FINSEQ_1:1;
    hence thesis by A1;
  end;
  suppose
    0 = n;
    then n in {0} by TARSKI:def 1;
    hence thesis by A2;
  end;
end;
