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 Th20:
  (r GeoSeq).(a+b) = (r GeoSeq).a * r|^b
  proof
    set S = r GeoSeq;
    defpred P[Nat] means S.(a+$1) = S.a * r|^$1;
A1: P[0]
    proof
      r|^0 = 1 by NEWTON:4;
      hence thesis;
    end;
A2: for k st P[k] holds P[k+1]
    proof
      let k such that
A3:   P[k];
      a+(k+1) = a+k+1;
      hence S.(a+(k+1)) = S.(a+k) * r|^1 by PREPOWER:3
      .= S.a * (r|^k * r) by A3
      .= S.a * r|^(k+1) by NEWTON:6;
    end;
    for k holds P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
