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

theorem Th43:
  for c being Element of k-chain-space(p),
      x being Element of (k-1)-polytopes(p)
  holds (Boundary(c))@x = Sum incidence-sequence(x,c)
proof
  let c be Element of k-chain-space(p), x be Element of (k-1)-polytopes(p);
  set b = Boundary(c);
  per cases;
  suppose
A1: (k-1)-polytopes(p) is empty;
    set iscx = incidence-sequence(x,c);
    iscx = <*>(the carrier of Z_2) by A1,Def16;
    then
A2: Sum iscx = 0.Z_2 by RLVECT_1:43;
    Boundary(c) = 0.((k-1)-chain-space(p)) by A1;
    hence thesis by A2,BSPACE:14;
  end;
  suppose
A3: (k-1)-polytopes(p) is non empty;
    then
A4: x in b iff Sum incidence-sequence(x,c) = 1.Z_2 by Def17;
    per cases;
    suppose
      x in b;
      hence thesis by A4,BSPACE:def 3;
    end;
    suppose
A5:   not x in b;
      then Sum incidence-sequence(x,c) <> 1.Z_2 by A3,Def17;
      then Sum incidence-sequence(x,c) = 0.Z_2 by BSPACE:8;
      hence thesis by A5,BSPACE:def 3;
    end;
  end;
end;
