 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem Telescoping:
  for a,b,s being Real_Sequence st
  (for n being Nat holds s.n = a.n + b.n) &
  (for k being Nat holds b.k = -(a.(k+1))) holds
    for n being Nat holds (Partial_Sums s).n = (a.0) + (b.n)
  proof
    let a,b,s be Real_Sequence;
    assume that
Z1: (for n being Nat holds s.n = a.n + b.n) and
Z2: (for k being Nat holds b.k = -(a.(k+1)));
    let n be Nat;
    defpred P[Nat] means
      (Partial_Sums s).$1 = (a.0) + (b.$1);
    (Partial_Sums s).0 = s.0 by SERIES_1:def 1
       .= (a.0) + (b.0) by Z1; then
A1: P[0];
A2: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k]; then
      (Partial_Sums s).(k+1) = (a.0) + (b.k) + s.(k + 1) by SERIES_1:def 1
         .= (a.0) + (b.k) + (a.(k + 1) + b.(k + 1)) by Z1
         .= (a.0) + (-a.(k+1)) + (a.(k + 1) + b.(k + 1)) by Z2
         .= (a.0) + b.(k + 1);
      hence thesis;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
