reserve n for Nat,
  k for Integer;
reserve p for polyhedron,
  k for Integer,
  n for Nat;

theorem Th23:
  k-polytopes(p) is non empty iff -1 <= k & k <= dim(p)
proof
  set X = k-polytopes(p);
  thus X is non empty implies -1 <= k & k <= dim(p) by Def5;
  thus -1 <= k & k <= dim(p) implies k-polytopes(p) is non empty
  proof
    assume
A1: -1 <= k;
    assume
A2: k <= dim(p);
    per cases by A1,A2,XXREAL_0:1;
    suppose
      k = -1;
      hence thesis by Def5;
    end;
    suppose
A3:   -1 < k & k < dim(p);
      set F = the PolytopsF of p;
A4:   k-polytopes(p) = rng (F.(k+1)) by A3,Def5;
      set n = k + 1;
A5:   1 <= n by A3,Th22;
A6:   n <= dim(p) by A3,Th22;
      reconsider n as Element of NAT by A5,INT_1:3;
      reconsider n as Nat;
      F.n is non empty by A5,A6,Def3;
      hence thesis by A4;
    end;
    suppose
      k = dim(p);
      then k-polytopes(p) = {p} by Def5;
      hence thesis;
    end;
  end;
end;
