
theorem Th3:
  for k being Element of NAT, X being finite non empty Subset of [:
NAT,{k}:] holds ex n being non zero Element of NAT st X c= [:Seg n \/ {0},{k}
  :]
proof
  let k be Element of NAT;
  let X be finite non empty Subset of [:NAT,{k}:];
  reconsider pX = proj1 X as finite non empty Subset of NAT by FUNCT_5:11;
  reconsider mpX = max pX as Element of NAT by ORDINAL1:def 12;
  reconsider m = mpX + 1 as non zero Element of NAT;
  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 consider x1,x2 being object such that
A4: x1 in NAT and
A5: x2 in {k} and
A6: x = [x1,x2] by ZFMISC_1:def 2;
  reconsider n = x`1 as Element of NAT by A4,A6;
  n in pX by A3,A6,XTUPLE_0:def 12;
  then max pX <= m & n <= max pX by NAT_1:11,XXREAL_2:def 8;
  then
A7: n <= m by XXREAL_0:2;
  per cases by NAT_1:25;
  suppose
    1 <= n;
    then n in Seg m by A7,FINSEQ_1:1;
    hence thesis by A5,A6,A1,ZFMISC_1:87;
  end;
  suppose
    0 = n;
    then n in {0} by TARSKI:def 1;
    hence thesis by A5,A6,A2,ZFMISC_1:87;
  end;
end;
