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

theorem
  (for n holds s.n = n |^ 4) implies for n holds Partial_Sums(s).n = n*(
  n+1)*(2*n+1)*(3*n|^2+3*n-1)/30
proof
  defpred X[Nat] means Partial_Sums(s).$1=$1*($1+1)*(2*$1+1)*(3*$1
  |^2+3*$1-1)/30;
  assume
A1: for n holds s.n = n |^ 4;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = n*(n+1)*(2*n+1)*(3*n|^2+3*n-1)/30;
    then
    Partial_Sums(s).(n+1) =n*(n+1)*(2*n+1)*(3*n|^2+3*n-1)/30 + s.(n+1) by
SERIES_1:def 1
      .=n*(n+1)*(2*n+1)*(3*n|^2+3*n-1)/30 + (n+1)|^4 by A1
      .=(n*(n+1)*(2*n+1)*(3*n|^2+3*n-1)+ (n+1)|^(3+1)*30)/30
      .=(n*(n+1)*(2*n+1)*(3*n|^2+3*n-1)+ (n+1)|^3*(n+1)*30)/30 by NEWTON:6
      .=(n+1)*(n*(2*n+1)*(3*n|^2+3*n-1)+ (n+1)|^3*30)/30
      .=(n+1)*((n+2)*(2*(n+1)+1)*(3*(n+1)|^2+3*(n+1)-1))/30 by Lm5;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0 |^4 by A1
    .=0*(0+1)*(2*0+1)*(3*0|^2+3*0-1)/30 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
