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

theorem Th31:
  k-polytope-seq(p) is one-to-one
proof
  set s = k-polytope-seq(p);
  per cases by XXREAL_0:1;
  suppose
    k < -1;
    hence thesis by Def7;
  end;
  suppose
    k = -1;
    hence thesis by Def7;
  end;
  suppose
A1: -1 < k & k < dim(p);
    then
A2: -1 + 1 < k + 1 by XREAL_1:6;
    then reconsider m = k + 1 as Element of NAT by INT_1:3;
    set F = the PolytopsF of p;
A3: 0 + 1 <= m by A2,INT_1:7;
    s = F.(k+1) & m <= dim(p) by A1,Def7,INT_1:7;
    hence thesis by A3,Def3;
  end;
  suppose
    k = dim(p);
    then s = <*p*> by Def7;
    hence thesis;
  end;
  suppose
    k > dim(p);
    hence thesis by Def7;
  end;
end;
