reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th24:
  (Partial_Sums(r GeoSeq)).(k+i+1) - (Partial_Sums(r GeoSeq)).k
  = r|^(k+1) * (Partial_Sums(r GeoSeq)).i
  proof
    set S = r GeoSeq;
    set P = Partial_Sums(S);
    defpred P[Nat] means P.(k+$1+1) - P.k = r|^(k+1) * P.$1;
A1: P[0]
    proof
A2:   P.0 = (r GeoSeq).0 by SERIES_1:def 1
      .= 1 by PREPOWER:3;
      P.(k+1) = P.k + S.(k+1) by SERIES_1:def 1;
      hence thesis by A2,PREPOWER:def 1;
    end;
A3: for a st P[a] holds P[a+1]
    proof
      let a such that
A4:   P[a];
A5:   P.(a+1) = P.a + S.(a+1) by SERIES_1:def 1;
A6:   S.(k+(a+1)) = S.(a+1) * r|^k by Th20;
A7:   S.(k+(a+1)+1) = S.(k+(a+1)) * r by PREPOWER:3
      .= S.(a+1) * (r|^k * r) by A6
      .= r|^(k+1) * S.(a+1) by NEWTON:6;
      P.(k+(a+1)+1) = P.(k+(a+1)) + S.(k+(a+1)+1) by SERIES_1:def 1;
      hence P.(k+(a+1)+1) - P.k = P.(k+(a+1)) - P.k + S.(k+(a+1)+1)
      .= r|^(k+1) * P.a + S.(k+(a+1)+1) by A4
      .= r|^(k+1) * P.(a+1) by A5,A7;
    end;
    for k holds P[k] from NAT_1:sch 2(A1,A3);
    hence thesis;
  end;
