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
  1<=k & k <=n or k=n iff n block k qua Nat > 0
proof
  thus 1<=k & k <=n or k=n implies n block k > 0
  proof
    set F={g where g is Function of Segm n,Segm k:g is onto "increasing};
    assume that
A1: 1<=k & k <=n or k=n;
    k=0 iff n=0 by A1;
    then consider f such that
A2: f is onto "increasing by A1,Th23;
    f in F by A2;
    hence thesis;
  end;
  thus n block k > 0 implies 1<=k & k <=n or k=n
  proof
    assume
A3: n block k >0;
    assume
A4: not ( 1<=k & k <=n or k=n);
    then 1+0>k or k>n;
    then k=0 & n<>k or k>n by A4,NAT_1:13;
    hence contradiction by A3,Th29,Th31;
  end;
end;
