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 Th25:
  n+1,m+1 are_coprime implies
  (Partial_Sums(a GeoSeq)).n, (Partial_Sums(a GeoSeq)).m are_coprime
  proof
    assume
A1: n+1,m+1 are_coprime;
A2: Partial_Sums(a GeoSeq).0 = (a GeoSeq).0 by SERIES_1:def 1
    .= 1 by PREPOWER:3;
    n+1 <> m+1 or n+1 = m+1 = 1 by A1,Th14;
    then per cases by XXREAL_0:1;
    suppose n < m or m < n;
      hence thesis by A1,Lm31;
    end;
    suppose n = m = 0;
      hence thesis by A2,NAT_D:32;
    end;
  end;
