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 Th21:
  for p,m being Integer st p divides m holds
  p divides (Partial_Sums (m GeoSeq)).n - 1
  proof
    let p,m be Integer such that
A1: p divides m;
    set G = m GeoSeq;
    set P = Partial_Sums G;
    defpred P[Nat] means p divides P.$1 - 1;
    P.0 = G.0 by SERIES_1:def 1
    .= 1 by PREPOWER:3;
    then
A2: P[0] by INT_2:12;
A3: for k st P[k] holds P[k+1]
    proof
      let k such that
A4:   P[k];
A5:   P.(k+1) = P.k + G.(k+1) by SERIES_1:def 1;
      p divides G.(k+1) by A1,Th19;
      then p divides P.k - 1 + G.(k+1) by A4,WSIERP_1:4;
      hence P[k+1] by A5;
    end;
    for k holds P[k] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
