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

theorem Th56:
  for k being Integer, x being Element of k-polytopes(p), v being
Element of k-chain-space(p), e being Element of (k-1)-polytopes(p), n being Nat
  st k = 0 & v = {x} & e = {} & x = n-th-polytope(p,k) & 1 <= n & n <=
  num-polytopes(p,k) holds incidence-sequence(e,v).n = 1.Z_2
proof
  let k be Integer, x be Element of k-polytopes(p), v be Element of k
  -chain-space(p), e be Element of (k-1)-polytopes(p), n be Nat such that
A1: k = 0 and
A2: v = {x} and
A3: e = {} and
A4: x = n-th-polytope(p,k) & 1 <= n & n <= num-polytopes(p,k);
  set iseq = incidence-sequence(e,v);
A5: x in v by A2,TARSKI:def 1;
  (k-1)-polytopes(p) is non empty by A1,Def5;
  then iseq.n = (v@x)*incidence-value(e,x) by A4,Def16
    .= (1.Z_2)*incidence-value(e,x) by A5,BSPACE:def 3
    .= (1.Z_2)*(1.Z_2) by A1,A3,Th55
    .= 1.Z_2;
  hence thesis;
end;
