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
  (n=0 iff k=0) implies
    ex f be Function of Segm n, Segm k st f is "increasing
proof
  assume
A1: n=0 iff k=0;
  now
    per cases;
    suppose
A2:   n=0;
      set f={};
A3:   dom f = n by A2;
      rng f= k by A1,A2;
      then reconsider f as Function of Segm n,Segm k by A3,FUNCT_2:1;
      f is "increasing by A1;
      hence thesis;
    end;
    suppose
      n>0;
      then consider f be Function such that
A4:   dom f=n and
A5:   rng f={0} by FUNCT_1:5;
      reconsider f as Function of n,{0} by A4,A5,FUNCT_2:1;
      rng f c= Segm k
      proof
        let x be object such that
A6:     x in rng f;
        x=0 by A6,TARSKI:def 1;
        hence thesis by A1,A6,NAT_1:44;
      end;
      then reconsider f as Function of Segm n,Segm k by FUNCT_2:6;
   for l,m st l in rng f & m in rng f & l < m holds min* f"{l} < min*
      f"{m} by A5,TARSKI:def 1;
      then f is "increasing by A1;
      hence thesis;
    end;
  end;
  hence thesis;
end;
