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

theorem Th41:
  for c,d being Element of k-chain-space(p) holds c = d iff for x
  being Element of k-polytopes(p) holds c@x = d@x
proof
  let c,d be Element of k-chain-space(p);
  thus c = d implies for x being Element of k-polytopes(p) holds c@x = d@x;
  thus (for x being Element of k-polytopes(p) holds c@x = d@x) implies c = d
  proof
    assume
A1: for x being Element of k-polytopes(p) holds c@x = d@x;
    thus c c= d
    proof
      let x be object such that
A2:   x in c;
      reconsider x as Element of k-polytopes(p) by A2;
      reconsider c as Subset of k-polytopes(p);
      c@x = 1.Z_2 by A2,BSPACE:def 3;
      then d@x = 1.Z_2 by A1;
      hence thesis by BSPACE:9;
    end;
    thus d c= c
    proof
      let x be object such that
A3:   x in d;
      reconsider x as Element of k-polytopes(p) by A3;
      reconsider d as Subset of k-polytopes(p);
      d@x = 1.Z_2 by A3,BSPACE:def 3;
      then c@x = 1.Z_2 by A1;
      hence thesis by BSPACE:9;
    end;
  end;
end;
