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

theorem Th11:
  for s,s1 st for n holds s1.n=s.0 holds Partial_Sums(s^\1) = (
  Partial_Sums(s)^\1) - s1
proof
  let s,s1;
  assume
A1: for n holds s1.n=s.0;
A2: now
    let k;
    thus ((Partial_Sums(s)^\1) - s1).(k+1) = (Partial_Sums(s)^\1).(k+1) - s1.(
    k+1) by RFUNCT_2:1
      .= (Partial_Sums(s)^\1).(k+1) - s.0 by A1
      .= Partial_Sums(s).(k+1+1) - s.0 by NAT_1:def 3
      .= s.(k+1+1) + Partial_Sums(s).(k+1) - s.0 by Def1
      .= s.(k+1+1) + Partial_Sums(s).(k+1) - s1.k by A1
      .= s.(k+1+1) + (Partial_Sums(s).(k+1) - s1.k)
      .= s.(k+1+1) + ((Partial_Sums(s)^\1).k - s1.k) by NAT_1:def 3
      .= s.(k+1+1) + ((Partial_Sums(s)^\1) - s1).k by RFUNCT_2:1
      .= ((Partial_Sums(s)^\1) - s1).k + (s^\1).(k+1) by NAT_1:def 3;
  end;
  ((Partial_Sums(s)^\1) - s1).0 = (Partial_Sums(s)^\1).0 - s1.0 by RFUNCT_2:1
    .= (Partial_Sums(s)^\1).0 - s.0 by A1
    .= Partial_Sums(s).(0+1) - s.0 by NAT_1:def 3
    .= Partial_Sums(s).0 + s.(0+1) - s.0 by Def1
    .= s.(0+1) + s.0 - s.0 by Def1
    .= (s^\1).0 by NAT_1:def 3;
  hence thesis by A2,Def1;
end;
