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

theorem Th50:
  alternating-semi-proper-f-vector(p) =
  alternating-proper-f-vector(p) ^ <*(-1)|^(dim(p))*>
proof
  set d = dim(p);
  set aspcs = alternating-semi-proper-f-vector(p);
  set apcs = alternating-proper-f-vector(p);
  set r = apcs ^ <*(-1)|^(dim(p))*>;
A1: len aspcs = d + 1 by Def28;
A2: for n being Nat st 1 <= n & n <= len aspcs holds aspcs.n = r.n
  proof
    let n be Nat such that
A3: 1 <= n and
A4: n <= len aspcs;
    per cases by A1,A4,NAT_1:8;
    suppose
A5:   n <= d;
      len apcs = d & dom apcs = Seg (len apcs) by Def27,FINSEQ_1:def 3;
      then n in dom apcs by A3,A5;
      then r.n = apcs.n by FINSEQ_1:def 7
        .= ((-1)|^(n+1))*num-polytopes(p,n-1) by A3,A5,Def27;
      hence thesis by A1,A3,A4,Def28;
    end;
    suppose
A6:   n = d + 1;
      then n = (len apcs) + 1 by Def27;
      then
A7:   r.n = (-1)|^d by FINSEQ_1:42
        .= (-1)|^(d+2) by Th12;
      aspcs.n = ((-1)|^(n+1))*num-polytopes(p,n-1) by A3,A6,Def28
        .= ((-1)|^(n+1))*1 by A6,Th29
        .= (-1)|^(n+1);
      hence thesis by A6,A7;
    end;
  end;
  len r = (len apcs) + (len <*(-1)|^(dim(p))*>) by FINSEQ_1:22
    .= d + (len <*(-1)|^(dim(p))*>) by Def27
    .= d + 1 by FINSEQ_1:40;
  then len aspcs = len r by Def28;
  hence thesis by A2;
end;
