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

theorem
  (for n holds s.n = 1/((2 -Root (n+1))+(2 -Root n))) implies
  Partial_Sums(s).n = 2 -Root (n+1)
proof
  defpred X[Nat] means Partial_Sums(s).$1 = 2-Root ($1+1);
  assume
A1: for n holds s.n = 1/((2 -Root (n+1))+(2 -Root n));
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = 2 -Root (n+1);
    then Partial_Sums(s).(n+1) = 2 -Root (n+1)+s.(n+1) by SERIES_1:def 1
      .=2 -Root (n+1)+1/((2 -Root ((n+1)+1))+(2 -Root (n+1))) by A1
      .=2 -Root (n+1)+((2 -Root (n+2))-(2 -Root (n+1))) by Lm7
      .=2 -Root (n+2);
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=1/((2 -Root (0+1))+(2 -Root 0)) by A1
    .=1/((2 -Root (0+1))+0) by PREPOWER:def 2
    .=1/1 by PREPOWER:20
    .=2 -Root (0+1) by PREPOWER:20;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
