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

theorem Th46:
  k-boundary(p) is linear-transformation of k-chain-space(p),(k-1)
  -chain-space(p)
proof
  set V = k-chain-space(p);
  set b = k-boundary(p);
A1: for a being Element of Z_2, x being Element of V holds b.(a*x) = a*(b.x)
  proof
    let a be Element of Z_2, x be Element of V;
    b.(a*x) = Boundary(a*x) by Def18
      .= a*(Boundary(x)) by Th45
      .= a*(b.x) by Def18;
    hence thesis;
  end;
  for x,y being Element of V holds b.(x+y) = (b.x) + (b.y)
  proof
    let x,y be Element of V;
    b.(x+y) = Boundary(x+y) by Def18
      .= Boundary(x) + Boundary(y) by Th44
      .= (b.x) + Boundary(y) by Def18
      .= (b.x) + (b.y) by Def18;
    hence thesis;
  end;
   then b is additive homogeneous by A1,MOD_2:def 2,VECTSP_1:def 20;
  hence thesis;
end;
