reserve m,n,s for non zero Element of NAT;

theorem Th4:
  for n being Nat holds Sum(n|->0.Z_2) = 0.Z_2
proof
  let n be Nat;
  set x = n|->0.Z_2;
A1: len x = len x;
  now let k be Nat;
    assume k in dom x;
    then x/.k = x.k by PARTFUN1:def 6
    .= 0.Z_2 by BSPACE:5;
    hence x.k = x/.k + x/.k by BSPACE:5;
  end;
  then
A2:Sum(x) - Sum(x) = Sum(x) + Sum(x) - Sum(x) by A1,RLVECT_2:2;
  Sum(x) + Sum(x) - Sum(x) = Sum(x) + (Sum(x) - Sum(x)) by RLVECT_1:28
  .= Sum(x) + 0.Z_2 by RLVECT_1:15
  .= Sum(x) by BSPACE:5;
  hence thesis by A2,RLVECT_1:15;
end;
