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

theorem Th6:
  Partial_Sums(s1) - Partial_Sums(s2) = Partial_Sums(s1-s2)
proof
A1: now
    let n;
    thus (Partial_Sums(s1) - Partial_Sums(s2)).(n+1) = Partial_Sums(s1).(n+1)
    - Partial_Sums(s2).(n+1) by RFUNCT_2:1
      .= (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
      .= (s1-s2).(n+1)+Partial_Sums(s1).n-Partial_Sums(s2).n by RFUNCT_2:1
      .= (s1-s2).(n+1)+(Partial_Sums(s1).n-Partial_Sums(s2).n)
      .= (Partial_Sums(s1)-Partial_Sums(s2)).n+(s1-s2).(n+1) by RFUNCT_2:1;
  end;
  (Partial_Sums(s1) - Partial_Sums(s2)).0 = Partial_Sums(s1).0 -
  Partial_Sums(s2).0 by RFUNCT_2:1
    .= s1.0 - Partial_Sums(s2).0 by Def1
    .= s1.0 - s2.0 by Def1
    .= (s1-s2).0 by RFUNCT_2:1;
  hence thesis by A1,Def1;
end;
