reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

theorem
  (a <> 1 & for n holds s.n = n*a|^n) implies for n holds Partial_Sums(s
  ).n = a*(1-a|^n)/(1-a)|^2 - n*a|^(n+1)/(1-a)
proof
  assume that
A1: a <> 1 and
A2: for n holds s.n = n*a|^n;
  defpred X[Nat] means Partial_Sums(s).$1 =a*(1-a|^$1)/(1-a)|^2-$1*
  a|^($1+1)/(1-a);
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = a*(1-a|^n)/(1-a)|^2-n*a|^(n+1)/(1-a);
    then
    Partial_Sums(s).(n+1) =a*(1-a|^n)/(1-a)|^2-n*a|^(n+1)/(1-a)+s.(n+1) by
SERIES_1:def 1
      .=a*(1-a|^n)/(1-a)|^2-n*a|^(n+1)/(1-a)+(n+1)*a|^(n+1) by A2
      .=a*(1-a|^n)/(1-a)|^2-n*a|^(n+1)/(1-a)+n*a|^(n+1)+a|^(n+1)
      .=a*(1-a|^(n+1))/(1-a)|^2- (n+1)*a|^(n+2)/(1-a) by A1,Lm6;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0*a|^0 by A2
    .=(1-1)*a/(1-a)|^2-0*a|^(0+1)/(1-a)
    .=(1-a|^0)*a/(1-a)|^2-0*a|^(0+1)/(1-a) by NEWTON:4;
  then
A4: X[0];
  for n holds X[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
