reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;

theorem Th5:
  for seq,n holds {seq.k: n <= k} is non empty Subset of ExtREAL
proof
  let seq,n;
  deffunc F(Nat) = seq.$1;
  defpred P[Nat] means n <= $1;
  set Y = {F(k): P[k]};
A1: seq.n in Y;
  Y c= ExtREAL
   proof let x be object;
    assume x in Y;
     then consider k such that
A2:   F(k) = x and
  P[k];
    thus thesis by A2;
   end;
  hence thesis by A1;
end;
