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

theorem Th39:
  for c being Element of k-chain-space(p), a being Element of Z_2,
  x being Element of k-polytopes(p) holds (a*c)@x = a*(c@x)
proof
  let c be Element of k-chain-space(p), a be Element of Z_2, x be Element of k
  -polytopes(p);
  per cases by BSPACE:8;
  suppose
    a = 0.Z_2;
    then a*(c@x) = 0.Z_2 & a*c = 0.(k-chain-space(p)) by VECTSP_1:14;
    hence thesis by BSPACE:14;
  end;
  suppose
    a = 1.Z_2;
    hence thesis;
  end;
end;
