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

theorem Th59:
  for x being Element of 0-polytopes(p) holds (0-boundary(p)).({x} ) = {{}}
proof
  reconsider minusone = 0-1 as Integer;
  let x be Element of 0-polytopes(p);
  set T = 0-boundary(p);
  0-polytopes(p) is non empty by Th52;
  then reconsider v = {x} as Subset of 0-polytopes(p) by ZFMISC_1:31;
  (0-1)-polytopes(p) = {{}} by Def5;
  then reconsider null = {} as Element of (0-1)-polytopes(p)
  by TARSKI:def 1;
  reconsider v as Element of 0-chain-space(p);
  reconsider bv = Boundary(v) as Element of minusone-chain-space(p);
A1: bv c= {null}
  proof
A2: [#](minusone-chain-space(p)) = { {}, {{}} } by Th54;
    let y be object such that
A3: y in bv;
    per cases by A2,TARSKI:def 2;
    suppose
      bv = {};
      hence thesis by A3;
    end;
    suppose
      bv = {{}};
      hence thesis by A3;
    end;
  end;
  minusone-polytopes(p) is non empty by Def5;
  then null in bv iff Sum incidence-sequence(null,v) = 1.Z_2 by Def17;
  then
A4: {null} c= bv by Th58,ZFMISC_1:31;
  T.v = Boundary(v) by Def18;
  hence thesis by A4,A1;
end;
