reserve i,j,x,y for object,
  f,g for Function;
reserve T,T1 for finite Tree,
  t,p for Element of T,
  t1 for Element of T1;

theorem
  for n being Nat holds
  {k where k is Nat: k > n} is infinite
proof
  let n be Nat;
  set X = {k where k is Nat: k > n};
A1: X c= NAT
  proof
    let x be object;
    assume x in X;
    then ex k being Nat st x = k & k > n;
    hence thesis by ORDINAL1:def 12;
  end;
  n+1 > n+0 by XREAL_1:8; then
A2: n+1 in X;
  assume X is finite;
  then reconsider X as non empty finite Subset of NAT by A1,A2;
  set m = max X;
  m in X by XXREAL_2:def 8;
  then consider k being Nat such that
A3: m = k and
A4: k > n;
  k+1 > k+0 by XREAL_1:8;
  then k+1 > n by A4,XXREAL_0:2;
  then m+1 in X by A3;
  then m+1 <= m+0 by XXREAL_2:def 8;
  hence contradiction by XREAL_1:8;
end;
