reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th5:
  for X being non empty add-closed complex-membered set
  for s1,s2 being sequence of X holds
  Partial_Sums(s1) + Partial_Sums(s2) = Partial_Sums(s1+s2)
proof
  let X be non empty add-closed complex-membered set;
  let s1,s2 be sequence of X;
A1: now
    let n;
    thus (Partial_Sums(s1) + Partial_Sums(s2)).(n+1) = Partial_Sums(s1).(n+1)
    + Partial_Sums(s2).(n+1) by Lm1
      .= Partial_Sums(s1).n + s1.(n+1) + Partial_Sums(s2).(n+1) by Def1
      .= Partial_Sums(s1).n+s1.(n+1)+(s2.(n+1)+Partial_Sums(s2).n) by Def1
      .= Partial_Sums(s1).n+(s1.(n+1)+s2.(n+1))+Partial_Sums(s2).n
      .= Partial_Sums(s1).n+(s1+s2).(n+1)+Partial_Sums(s2).n by Lm1
      .= Partial_Sums(s1).n+Partial_Sums(s2).n+(s1+s2).(n+1)
      .= (Partial_Sums(s1)+Partial_Sums(s2)).n+(s1+s2).(n+1) by Lm1;
  end;
  (Partial_Sums(s1) + Partial_Sums(s2)).0 = Partial_Sums(s1).0 +
  Partial_Sums(s2).0 by Lm1
    .= s1.0 + Partial_Sums(s2).0 by Def1
    .= s1.0 + s2.0 by Def1
    .= (s1+s2).0 by Lm1;
  hence thesis by A1,Def1;
end;
