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

theorem Th44:
  for c,d being Element of k-chain-space(p) holds
  Boundary(c+d) = Boundary(c) + Boundary(d)
proof
  let c,d be Element of k-chain-space(p);
  set bc = Boundary(c);
  set bd = Boundary(d);
  set s = c+d;
  set l = Boundary(s);
  set r = bc+bd;
  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 a = bc@x;
    set b = bd@x;
A1: a = Sum incidence-sequence(x,c) & b = Sum incidence-sequence(x,d) by Th43;
    l@x = Sum incidence-sequence(x,s) & r@x = a+b by Th35,Th43;
    hence thesis by A1,Th38;
  end;
  hence thesis by Th41;
end;
