let p be polyhedron; :: thesis: for k being Integer
for x being Element of k -polytopes p
for v being Element of (k -chain-space p)
for e being Element of (k - 1) -polytopes p st k = 0 & v = {x} & e = {} holds
Sum (incidence-sequence e,v) = 1. Z_2
let k be Integer; :: thesis: for x being Element of k -polytopes p
for v being Element of (k -chain-space p)
for e being Element of (k - 1) -polytopes p st k = 0 & v = {x} & e = {} holds
Sum (incidence-sequence e,v) = 1. Z_2
let x be Element of k -polytopes p; :: thesis: for v being Element of (k -chain-space p)
for e being Element of (k - 1) -polytopes p st k = 0 & v = {x} & e = {} holds
Sum (incidence-sequence e,v) = 1. Z_2
let v be Element of (k -chain-space p); :: thesis: for e being Element of (k - 1) -polytopes p st k = 0 & v = {x} & e = {} holds
Sum (incidence-sequence e,v) = 1. Z_2
let e be Element of (k - 1) -polytopes p; :: thesis: ( k = 0 & v = {x} & e = {} implies Sum (incidence-sequence e,v) = 1. Z_2 )
assume that
A1: k = 0 and
A2: v = {x} and
A3: e = {} ; :: thesis: Sum (incidence-sequence e,v) = 1. Z_2
set iseq = incidence-sequence e,v;
( - 1 <= k & k <= dim p ) by A1;
then consider n being Nat such that
A4: x = n -th-polytope p,k and
A5: 1 <= n and
A6: n <= num-polytopes p,k by Th33;
not (k - 1) -polytopes p is empty by A1, Def5;
then A7: len (incidence-sequence e,v) = num-polytopes p,k by Def16;
dom (incidence-sequence e,v) = Seg (len (incidence-sequence e,v)) by FINSEQ_1:def 3;
then A8: n in dom (incidence-sequence e,v) by A5, A6, A7, FINSEQ_1:3;
A9: (incidence-sequence e,v) . n = 1. Z_2 by A1, A2, A3, A4, A5, A6, Th59;
for m being Nat st m in dom (incidence-sequence e,v) & m <> n holds
(incidence-sequence e,v) . m = 0. Z_2

proof

let m be Nat; :: thesis: ( m in dom (incidence-sequence e,v) & m <> n implies (incidence-sequence e,v) . m = 0. Z_2 )
assume that
A10: m in dom (incidence-sequence e,v) and
A11: m <> n ; :: thesis: (incidence-sequence e,v) . m = 0. Z_2
m in Seg (len (incidence-sequence e,v)) by A10, FINSEQ_1:def 3;
then ( 1 <= m & m <= len (incidence-sequence e,v) ) by FINSEQ_1:3;
hence (incidence-sequence e,v) . m = 0. Z_2 by A1, A2, A4, A5, A6, A7, A11, Th60; :: thesis: verum

end;

hence Sum (incidence-sequence e,v) = 1. Z_2 by A8, A9, MATRIX_3:14; :: thesis: verum