
theorem
  for n0 being Element of NAT ex NS being increasing sequence of NAT st
  for n being Element of NAT holds NS.n=n+n0
proof
  let n0 be Element of NAT;
  deffunc F(Nat) = $1 + n0;
  consider NS being Real_Sequence such that
A1: for n being Nat holds NS.n=F(n) from SEQ_1:sch 1;
A2: NS is increasing
  proof
    let n be Nat;
    n < n + 1 by NAT_1:13;
    then n + n0 < (n+1) + n0 by XREAL_1:6;
    then NS.n < (n+1) + n0 by A1;
    hence thesis by A1;
  end;
  for n being Nat holds NS.n is Element of NAT
  proof
    let n be Nat;
    n + n0 in NAT;
    hence thesis by A1;
  end;
  then reconsider NS as increasing sequence of NAT by A2,SEQM_3:13;
  take NS;
  thus thesis by A1;
end;
