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

theorem Th60:
  k = -1 implies dim(k-bounding-chain-space(p)) = 1
proof
  set T = 0-boundary(p);
  set V = k-bounding-chain-space(p);
  assume
A1: k = -1;
  card [#]V = 2
  proof
    [#]V c= [#](k-chain-space(p)) by VECTSP_4:def 2;
    then card [#]V c= card [#](k-chain-space(p)) by CARD_1:11;
    then
A2: card [#]V c= 2 by A1,Th53;
    0-polytopes(p) <> {} by Th52;
    then consider x being object such that
A3: x in 0-polytopes(p) by XBOOLE_0:def 1;
    reconsider x as Element of 0-polytopes(p) by A3;
    set v = {x};
A4: T.v = {{}} by Th59;
    reconsider v as Subset of 0-polytopes(p) by A3,ZFMISC_1:31;
    reconsider v as Element of 0-chain-space(p);
A5: dom T = [#](0-chain-space(p)) by RANKNULL:7;
    then v in dom T;
    then
A6: {{}} in rng T by A4,FUNCT_1:3;
    T.(0.(0-chain-space(p))) = 0.(k-chain-space(p)) by A1,RANKNULL:9
      .= {};
    then {} in rng T by A5,FUNCT_1:3;
    then
A7: {{},{{}}} c= rng T by A6,ZFMISC_1:32;
    card {{},{{}}} = 2 by CARD_2:57;
    then
A8: 2 c= card rng T by A7,CARD_1:11;
    card rng T = card (T .: [#](0-chain-space(p))) by RELSET_1:22
      .= card [#]V by A1,RANKNULL:def 2;
    hence thesis by A8,A2;
  end;
  hence thesis by RANKNULL:6;
end;
