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

theorem
  (for n holds s.n = 10|^n+2*n+1) implies Partial_Sums(s).n = 10|^(n+1)/
  9-1/9+(n+1)|^2
proof
  defpred X[Nat] means Partial_Sums(s).$1 =10|^($1+1)/9-1/9+($1+1)|^2;
  assume
A1: for n holds s.n = 10|^n+2*n+1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = 10|^(n+1)/9-1/9+(n+1)|^2;
    then Partial_Sums(s).(n+1) = 10|^(n+1)/9-1/9+(n+1)|^2 + s.(n+1) by
SERIES_1:def 1
      .= 10|^(n+1)/9-1/9+(n+1)|^2+(10|^(n+1)+2*(n+1)+1) by A1
      .= (10|^(n+1)*10)/9-1/9+((n+1)|^2+2*n+3)
      .= (10|^(n+1)*10)/9-1/9+(n|^2+2*n*1+1|^2+2*n+3) by Lm1
      .= (10|^(n+1)*10)/9-1/9+(n|^2+2*n+1+2*n+3)
      .= ((10|^(n+1)*10)/9-1/9)+(n|^2+2*n*2+2|^2) by Lm1
      .= ((10|^(n+1)*10)/9-1/9)+(n+2)|^2 by Lm1
      .= 10|^(n+1+1)/9-1/9+(n+2)|^2 by NEWTON:6;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=10|^0+2*0+1 by A1
    .=10/9-1/9+1 by NEWTON:4
    .=10|^(0+1)/9-1/9+1
    .=10|^(0+1)/9-1/9+(0+1)|^2;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
