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

theorem Th42:
  for c,d being Element of k-chain-space(p) holds c = d iff for x
  being Element of k-polytopes(p) holds x in c iff x in d
proof
  let c,d be Element of k-chain-space(p);
  thus c = d implies for x being Element of k-polytopes(p) holds x in c iff x
  in d;
  thus (for x being Element of k-polytopes(p) holds x in c iff x in d) implies
  c = d
  proof
    assume
A1: for x being Element of k-polytopes(p) holds x in c iff x in d;
    assume c <> d;
    then consider x being Element of k-polytopes(p) such that
A2: c@x <> d@x by Th41;
    not (x in c iff x in d) by A2,BSPACE:13;
    hence thesis by A1;
  end;
end;
