
theorem Th4:
  for m being Element of NAT, X being finite non empty Subset of [:
  NAT,{m}:] holds ex k being non zero Element of NAT st not [2*k+1,m] in X
proof
  let m be Element of NAT;
  let X be finite non empty Subset of [:NAT,{m}:];
  consider n being non zero Element of NAT such that
A1: X c= [:Seg n \/ {0},{m}:] by Th3;
A2: not [2*n+1,m] in X
  proof
    assume [2*n+1,m] in X;
    then
A3: 2*n+1 in Seg n \/ {0} by A1,ZFMISC_1:87;
    not 2*n+1 in {0} by TARSKI:def 1;
    then 2*n+1 in Seg n by A3,XBOOLE_0:def 3;
    then
A4: 2*n+1 <= n by FINSEQ_1:1;
    1*n <= 2*n by NAT_1:4;
    hence thesis by A4,NAT_1:13;
  end;
  assume for k being non zero Element of NAT holds [2*k+1,m] in X;
  hence contradiction by A2;
end;
