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

theorem Th74:
  (dim(p)-boundary(p)).{p} = (dim(p)-1)-polytopes(p)
proof
  reconsider c = {p} as Element of dim(p)-chain-space(p) by Th62;
  set T = dim(p)-boundary(p);
  set X = (dim(p)-1)-polytopes(p);
  reconsider d = X as Element of (dim(p)-1)-chain-space(p) by ZFMISC_1:def 1;
  reconsider Tc = T.c as Element of (dim(p)-1)-chain-space(p);
  for x being Element of X holds x in Tc iff x in d
  proof
    let x be Element of X;
    thus x in Tc implies x in d;
    thus x in d implies x in Tc
    proof
      assume x in d;
      Sum incidence-sequence(x,c) = 1.Z_2 by Th73;
      then x in Boundary(c) by Def17;
      hence thesis by Def18;
    end;
  end;
  hence thesis by SUBSET_1:3;
end;
