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

theorem Th40:
  for c being Element of k-chain-space(p), a being Element of Z_2,
  x being Element of (k-1)-polytopes(p) holds incidence-sequence(x,a*c) = a*
  incidence-sequence(x,c)
proof
  let c be Element of k-chain-space(p), a be Element of Z_2, x be Element of (
  k-1)-polytopes(p);
  set l = incidence-sequence(x,a*c);
  set isc = incidence-sequence(x,c);
  set r = a*isc;
  per cases;
  suppose
A1: (k-1)-polytopes(p) is empty;
    then isc = <*>(the carrier of Z_2) by Def16;
    then reconsider isc as Element of 0-tuples_on the carrier of Z_2 by
FINSEQ_2:131;
    a*isc is Element of 0-tuples_on the carrier of Z_2;
    then reconsider r as Element of 0-tuples_on the carrier of Z_2;
    r = <*>(the carrier of Z_2);
    hence thesis by A1,Def16;
  end;
  suppose
A2: (k-1)-polytopes(p) is non empty;
    set n = num-polytopes(p,k);
A3: len l = n & len r = n
    proof
      len isc = n by A2,Def16;
      then reconsider isc as Element of n-tuples_on the carrier of Z_2 by
FINSEQ_2:92;
      set r = a*isc;
      reconsider r as Element of n-tuples_on the carrier of Z_2;
      len r = n by CARD_1:def 7;
      hence thesis by A2,Def16;
    end;
    for m being Nat st 1 <= m & m <= len l holds l.m = r.m
    proof
A4:   dom r = Seg n by A3,FINSEQ_1:def 3;
      let m be Nat such that
A5:   1 <= m & m <= len l;
      set s = m-th-polytope(p,k);
      set ivs = incidence-value(x,s);
A6:   len l = n by A2,Def16;
      then
A7:   l.m = ((a*c)@s)*ivs by A2,A5,Def16;
      len l = n & m in NAT by A2,Def16,ORDINAL1:def 12;
      then
A8:   m in Seg n by A5;
      isc.m = (c@s)*ivs by A2,A5,A6,Def16;
      then r.m = a*((c@s)*ivs) by A4,A8,FVSUM_1:50
        .= (a*(c@s))*ivs by GROUP_1:def 3
        .= l.m by A7,Th39;
      hence thesis;
    end;
    hence thesis by A3;
  end;
end;
