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 Th23:
  (Partial_Sums(a GeoSeq)).k gcd (Partial_Sums(a GeoSeq)).(k+i)
  = (Partial_Sums(a GeoSeq)).k
  gcd ( (Partial_Sums(a GeoSeq)).(k+i) - (Partial_Sums(a GeoSeq)).k )
  proof
    set P = Partial_Sums(a GeoSeq);
    reconsider N = P.(k+i) - P.k as Element of NAT by INT_1:5,SEQM_3:5;
    P.(k+i) = P.k + N;
    hence thesis by FIB_NUM:1;
  end;
