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 Th22:
  (Partial_Sums (m GeoSeq)).n, m|^(n+1) are_coprime
  proof
    set P = (Partial_Sums (m GeoSeq)).n;
    assume not thesis;
    then consider p being Prime such that
A1: p divides P and
A2: p divides m|^(n+1) by PYTHTRIP:def 2;
    p divides P - 1 by A2,Th21,NAT_3:5;
    then p divides P - (P-1) by A1,INT_5:1;
    then p = 1 by WSIERP_1:15;
    hence thesis by INT_2:def 4;
  end;
