reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th7:
  for q,q1,q2 being FinSequence of REAL st len q1 = len q & len q1
= len q2 & (for k st k in dom q1 holds q.k = q1.k + q2.k) holds Sum q = Sum q1
  + Sum q2
proof
  let q,q1,q2 be FinSequence of REAL such that
A1: len q1 = len q and
A2: len q1 = len q2 and
A3: for k st k in dom q1 holds q.k = q1.k + q2.k;
  for k being Element of NAT st k in dom q1 holds q.k = q1/.k + q2/.k
  proof
    let k be Element of NAT such that
A4: k in dom q1;
A5: k in dom q2 by A2,A4,FINSEQ_3:29;
    thus q.k = q1.k + q2.k by A3,A4
      .= q1/.k + q2.k by A4,PARTFUN1:def 6
      .= q1/.k + q2/.k by A5,PARTFUN1:def 6;
  end;
  hence thesis by A1,A2,INTEGRA1:21;
end;
