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 Th19:
  n <> 0 implies
  for p,m being Integer st p divides m holds p divides (m GeoSeq).n
  proof
    assume
A1: n <> 0;
    let p,m be Integer such that
A2: p divides m;
    set G = m GeoSeq;
    defpred P[Nat] means $1 <> 0 implies p divides G.$1;
A3: G.0 = 1 by PREPOWER:3;
    G.(0+1) = G.0 * m by PREPOWER:3;
    then
A4: P[1] by A2,A3;
A5: for k being non zero Nat st P[k] holds P[k+1]
    proof
      let k be non zero Nat;
      G.(k+1) = G.k * m by PREPOWER:3;
      hence thesis by INT_2:2;
    end;
    for k being non zero Nat holds P[k] from NAT_1:sch 10(A4,A5);
    hence thesis by A1;
  end;
