reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;

theorem Th18:
  f is onto "increasing implies for m st m < k holds m <= min* f"{ m}
proof
  defpred M[Nat] means $1<k implies $1<=min* f"{$1};
  assume that
A1: f is onto "increasing;
A2: for m st M[m] holds M[m + 1]
  proof
A3: k=rng f by A1,FUNCT_2:def 3;
    let m such that
A4: M[m];
    assume
A5: m+1<k;
    m<m+1 by NAT_1:19;
    then m < k by A5,XXREAL_0:2;
    then
A6: m in rng f by A3,NAT_1:44;
A7: m < m+1 by NAT_1:19;
    m+1 in rng f by A5,A3,NAT_1:44;
    then min* f"{m}< min* f"{m+1} by A1,A6,A7;
    then m < min* f"{m+1} by A4,A5,A7,XXREAL_0:2;
    hence thesis by NAT_1:13;
  end;
A8: M[0];
  for m holds M[m] from NAT_1:sch 2(A8,A2);
  hence thesis;
end;
