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

theorem Th45:
  for a being Element of Z_2, c being Element of k-chain-space(p)
  holds Boundary(a*c) = a*(Boundary(c))
proof
  let a be Element of Z_2, c be Element of k-chain-space(p);
  set lsm = a*c;
  set l = Boundary(lsm);
  set rb = Boundary(c);
  set r = a*rb;
  for x being Element of (k-1)-polytopes(p) holds l@x = r@x
  proof
    let x be Element of (k-1)-polytopes(p);
    set b = rb@x;
A1: l@x = Sum incidence-sequence(x,lsm) & rb@x = Sum incidence-sequence(x,
    c) by Th43;
    r@x = a*b & incidence-sequence(x,lsm) = a*incidence-sequence(x,c) by Th39
,Th40;
    hence thesis by A1,FVSUM_1:73;
  end;
  hence thesis by Th41;
end;
