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

theorem Th27:
  rng (k-polytope-seq(p)) = k-polytopes(p)
proof
  set F = the PolytopsF of p;
  per cases;
  suppose
A1: k < -1;
    then k-polytopes(p) = {} by Def5;
    hence thesis by A1,Def7,RELAT_1:38;
  end;
  suppose
A2: -1 <= k & k <= dim(p);
    per cases by A2,XXREAL_0:1;
    suppose
      k = -1;
      then k-polytopes(p) = {{}} & k-polytope-seq(p) = <*{}*> by Def5,Def7;
      hence thesis by FINSEQ_1:38;
    end;
    suppose
A3:   -1 < k & k < dim(p);
      then k-polytopes(p) = rng (F.(k+1)) by Def5;
      hence thesis by A3,Def7;
    end;
    suppose
      k = dim(p);
      then k-polytopes(p) = {p} & k-polytope-seq(p) = <*p*> by Def5,Def7;
      hence thesis by FINSEQ_1:38;
    end;
  end;
  suppose
A4: k > dim(p);
    then k-polytopes(p) = {} by Def5;
    hence thesis by A4,Def7,RELAT_1:38;
  end;
end;
