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

theorem
  (for n holds s.n = 2*n-1+(1/2)|^n) implies Partial_Sums(s).n =n|^2+1-(
  1/2)|^n
proof
  defpred X[Nat] means Partial_Sums(s).$1 =$1|^2+1-(1/2)|^$1;
  assume
A1: for n holds s.n = 2*n-1+(1/2)|^n;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = n|^2+1-(1/2)|^n;
    then Partial_Sums(s).(n+1) =n|^2+1-(1/2)|^n+ s.(n+1) by SERIES_1:def 1
      .=n|^2+1-(1/2)|^n+(2*(n+1)-1+(1/2)|^(n+1)) by A1
      .=n|^2+1-(1/2)|^n+(1/2)|^(n+1)+2*(n+1)-1
      .=n|^2+1-(1/2)|^n+(1/2)|^n*(1/2)+2*(n+1)-1 by NEWTON:6
      .=n|^2+1-((1/2)|^n*(1/2))+2*(n+1)-1
      .=n|^2+1-(1/2)|^(n+1)+2*(n+1)-1 by NEWTON:6
      .=(n|^2+2*n+1)+1-(1/2)|^(n+1)
      .=(n|^2+2*n*1+1|^2)+1-(1/2)|^(n+1)
      .=(n+1)|^2+1-(1/2)|^(n+1) by Lm1;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=2*0-1+(1/2)|^0 by A1
    .=-1+1 by NEWTON:4
    .=1-1+0|^2 by NEWTON:11
    .=1-(1/2)|^0+0|^2 by NEWTON:4;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
