reserve n,k,b for Nat, i for Integer;

theorem Th10:
  for cF being complex-valued XFinSequence
  for B1,B2 being set st
  B1 misses B2 holds
  Sum(SubXFinS(cF,B1\/B2))=Sum(SubXFinS(cF,B1))+Sum(SubXFinS(cF,B2))
  proof
    let cF be complex-valued XFinSequence;
    let B1,B2 be set such that A1: B1 misses B2;
    set O=SubXFinS(cF,B1\/B2);
    reconsider O=SubXFinS(cF,B1\/B2) as XFinSequence of COMPLEX;
    consider P be Permutation of dom O such that
    A2: O * P = SubXFinS (cF,B1) ^ SubXFinS (cF,B2) by A1,Th9;
    Sum(O * P) = Sum(SubXFinS(cF,B1)) + Sum(SubXFinS(cF,B2)) by A2,AFINSQ_2:55;
    hence thesis by AFINSQ_2:45;
  end;
