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

theorem Th78:
  1 < n & n < dim(p) + 2 implies (alternating-f-vector(p)).n = (
  alternating-proper-f-vector(p)).(n-1)
proof
  assume
A1: 1 < n;
  1 - 1 = 0;
  then reconsider m = n - 1 as Element of NAT by A1,INT_1:3,XREAL_1:13;
  reconsider m as Nat;
  set apcs = alternating-proper-f-vector(p);
  set acs = alternating-f-vector(p);
A2: 2 - 1 = 1;
  1 + 1 = 2;
  then 2 <= n by A1,INT_1:7;
  then
A3: 1 <= m by A2,XREAL_1:13;
  assume
A4: n < dim(p) + 2;
  then n < (dim(p) + 1) + 1;
  then n <= dim(p) + 1 by NAT_1:13;
  then n - 1 <= (dim(p) + 1) - 1 by XREAL_1:9;
  then
A5: apcs.m = ((-1)|^(m+1))*num-polytopes(p,m-1) by A3,Def27;
  acs.n = ((-1)|^n)*num-polytopes(p,n-2) by A1,A4,Def26;
  hence thesis by A5;
end;
