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

theorem Th36:
  for c,d being Element of k-chain-space(p), x being Element of (k
  -1)-polytopes(p) holds incidence-sequence(x,c+d) = incidence-sequence(x,c) +
  incidence-sequence(x,d)
proof
  let c,d be Element of k-chain-space(p), x be Element of (k-1)-polytopes(p);
  set n = num-polytopes(p,k);
  set l = incidence-sequence(x,c+d);
  set isc = incidence-sequence(x,c);
  set isd = incidence-sequence(x,d);
  set r = isc + isd;
  per cases;
  suppose
A1: (k-1)-polytopes(p) is empty;
    then isd is Tuple of 0, 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;
    isc + isd is Element of 0-tuples_on the carrier of Z_2;
    hence thesis by A1,Def16;
  end;
  suppose
A2: (k-1)-polytopes(p) is non empty;
A3: len(l) = n & len(r) = n
    proof
      len isd = n by A2,Def16;
      then reconsider isd as Element of n-tuples_on the carrier of Z_2 by
FINSEQ_2:92;
      len isc = n by A2,Def16;
      then reconsider isc as Element of n-tuples_on the carrier of Z_2 by
FINSEQ_2:92;
      reconsider s = isc + isd as Element of n-tuples_on the carrier of Z_2;
      len s = n by CARD_1:def 7;
      hence thesis by A2,Def16;
    end;
    for n being Nat st 1 <= n & n <= len l holds l.n = r.n
    proof
A4:   dom r = Seg n & len l = n by A3,FINSEQ_1:def 3;
      let m be Nat such that
A5:   1 <= m & m <= len l;
      set a = m-th-polytope(p,k);
      set iva = incidence-value(x,a);
A6:   len l = n by A2,Def16;
      then
A7:   l.m = ((c+d)@a)*iva by A2,A5,Def16;
A8:   m in dom r by A4,A5;
      isc.m = (c@a)*iva & isd.m = (d@a)*iva by A2,A5,A6,Def16;
      then r.m = (c@a)*iva + (d@a)*iva by A8,FVSUM_1:17
        .= (c@a + d@a)*iva by VECTSP_1:def 3
        .= l.m by A7,Th35;
      hence thesis;
    end;
    hence thesis by A3;
  end;
end;
