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 Th17:
  f is onto implies n>=k
proof
  assume
A1: f is onto;
  now
    per cases;
    suppose
      k=0;
      hence thesis;
    end;
    suppose
A2:   k<>0;
A3:   rng f = k by A1,FUNCT_2:def 3;
      dom f =n by A2,FUNCT_2:def 1;
      then Segm card k c= Segm card n by A3,CARD_1:12;
      then
A4:   card k <= card n by NAT_1:39;
      thus thesis by A4;
    end;
  end;
  hence thesis;
end;
