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

theorem Th55:
  for x being Element of k-polytopes(p), e being Element of (k-1)
  -polytopes(p) st k = 0 & e = {} holds incidence-value(e,x) = 1.Z_2
proof
  let x be Element of k-polytopes(p), e be Element of (k-1)-polytopes(p) such
  that
A1: k = 0 and
A2: e = {};
A3: eta(p,k) = [:{{}},0-polytopes(p):] --> 1.Z_2 by A1,Def6;
A4: k <= dim(p) by A1;
  then {} in {{}} & 0-polytopes(p) is non empty by Th23,TARSKI:def 1;
  then
A5: [{},x] in [:{{}},0-polytopes(p):] by A1,ZFMISC_1:87;
  incidence-value(e,x) = eta(p,k).(e,x) by A1,A4,Def13
    .= 1.Z_2 by A2,A3,A5,FUNCOP_1:7;
  hence thesis;
end;
