reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th1:
  for X being non empty natural-membered set st
   for a st a in X ex b st b > a & b in X
  holds X is infinite
  proof
    let X be non empty natural-membered set such that
A1: for a st a in X ex b st b > a & b in X;
    assume X is finite;
    then reconsider X as non empty finite ext-real-membered set;
    max X in X by XXREAL_2:def 8;
    then ex b st b > max X & b in X by A1;
    hence contradiction by XXREAL_2:def 8;
  end;
