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

theorem Th58:
  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) st k = 0 & v
  = {x} & e = {} holds Sum incidence-sequence(e,v) = 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) such that
A1: k = 0 and
A2: v = {x} and
A3: e = {};
  set iseq = incidence-sequence(e,v);
  k <= dim(p) by A1;
  then consider n being Nat such that
A4: x = n-th-polytope(p,k) and
A5: 1 <= n & n <= num-polytopes(p,k) by A1,Th30;
  (k-1)-polytopes(p) is non empty by A1,Def5;
  then
A6: len iseq = num-polytopes(p,k) by Def16;
A7: for m being Nat st m in dom iseq & m <> n holds iseq.m = 0.Z_2
  proof
    let m be Nat such that
A8: m in dom iseq and
A9: m <> n;
    m in Seg (len iseq) by A8,FINSEQ_1:def 3;
    then 1 <= m & m <= len iseq by FINSEQ_1:1;
    hence thesis by A1,A2,A4,A5,A6,A9,Th57;
  end;
  dom iseq = Seg (len iseq) by FINSEQ_1:def 3;
  then
A10: n in dom iseq by A5,A6;
  iseq.n = 1.Z_2 by A1,A2,A3,A4,A5,Th56;
  hence thesis by A10,A7,MATRIX_3:12;
end;
