reserve i,j for Nat;

theorem Th3:
  for F being FinSequence of REAL holds F + -F = 0*(len F) & F - F = 0*(len F)
proof
  let F be FinSequence of REAL;
  set n=len F;
  reconsider R=F as Element of n-tuples_on REAL by FINSEQ_2:92;
A1: R + -R = (n|->(0 qua Real)) by RVSUM_1:22;
  hence F + -F = 0*(len F) by EUCLID:def 4;
  thus thesis by A1,EUCLID:def 4;
end;
