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

theorem Th37:
  for c,d being Element of k-chain-space(p), x being Element of (k
-1)-polytopes(p) holds Sum (incidence-sequence(x,c) + incidence-sequence(x,d))
  = (Sum incidence-sequence(x,c)) + (Sum incidence-sequence(x,d))
proof
  let c,d be Element of k-chain-space(p), x be Element of (k-1)-polytopes(p);
  set isc = incidence-sequence(x,c);
  set isd = incidence-sequence(x,d);
  per cases;
  suppose
A1: (k-1)-polytopes(p) is empty;
    then isd = <*>(the carrier of Z_2) by Def16;
    then reconsider isd as Element of 0-tuples_on the carrier of Z_2 by
FINSEQ_2:131;
    isc = <*>(the carrier of Z_2) by A1,Def16;
    then reconsider isc as Element of 0-tuples_on the carrier of Z_2 by
FINSEQ_2:131;
    reconsider s = isc + isd as Element of 0-tuples_on the carrier of Z_2;
    Sum s = 0.Z_2 by FVSUM_1:74;
    hence thesis by RLVECT_1:def 4;
  end;
  suppose
A2: (k-1)-polytopes(p) is non empty;
    reconsider n = num-polytopes(p,k) as Element of NAT;
    len isd = n by A2,Def16;
    then reconsider
    isd9 = isd as Element of n-tuples_on the carrier of Z_2 by FINSEQ_2:92;
    len isc = n by A2,Def16;
    then reconsider
    isc9 = isc as Element of n-tuples_on the carrier of Z_2 by FINSEQ_2:92;
    Sum (isc + isd) = Sum (isc9 + isd9) .= Sum (isc) + Sum (isd) by FVSUM_1:76;
    hence thesis;
  end;
end;
