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

theorem
  (for n holds s.n = n|^6) implies for n holds Partial_Sums(s).n = n*(n+
  1)*(2*n+1)*(3*n|^4+6*n|^3-3*n+1)/42
proof
  defpred X[Nat] means Partial_Sums(s).$1=$1*($1+1)*(2*$1+1)*(3*$1
  |^4+6*$1|^3-3*$1+1)/42;
  assume
A1: for n holds s.n = n|^6;
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|^4+6*n|^3-3*n+1)/42;
    then
    Partial_Sums(s).(n+1) =n*(n+1)*(2*n+1)*(3*n|^4+6*n|^3-3*n+1)/42+s.(n+1
    ) by SERIES_1:def 1
      .=n*(n+1)*(2*n+1)*(3*n|^4+6*n|^3-3*n+1)/42+(n+1)|^6 by A1
      .=(n*(n+1)*(2*n+1)*(3*n|^4+6*n|^3-3*n+1)+(n+1)|^(5+1)*42)/42
      .=(n*(n+1)*(2*n+1)*(3*n|^4+6*n|^3-3*n+1)+(n+1)|^5*(n+1)*42)/42 by
NEWTON:6
      .=(n+1)*((2*(n*n)+n)*(3*n|^4+6*n|^3-3*n+1)+(n+1)|^5*42)/42
      .=(n+1)*((2*n|^2+n)*(3*n|^4+6*n|^3-3*n+1)+(n+1)|^5*42)/42 by WSIERP_1:1
      .=(n+1)*(3*(n|^4*n|^2)*2+3*(n|^4*n)+(6*(n|^3*n|^2)*2+6*(n|^3*n)) -(3*(
    n*n|^2)*2+3*(n*n))+(2*n|^2+n)+(n+1)|^5*42)/42
      .=(n+1)*(3*n|^(4+2)*2+3*(n|^4*n)+(6*(n|^3*n|^2)*2+6*(n|^3*n)) -(3*(n*n
    |^2)*2+3*(n*n))+(2*n|^2+n)+(n+1)|^5*42)/42 by NEWTON:8
      .=(n+1)*(3*n|^(4+2)*2+3*(n|^4*n)+(6*n|^(3+2)*2+6*(n|^3*n)) -(3*(n*n|^2
    )*2+3*(n*n))+(2*n|^2+n)+(n+1)|^5*42)/42 by NEWTON:8
      .=(n+1)*(3*n|^(4+2)*2+3*n|^(4+1)+(6*n|^(3+2)*2+6*(n|^3*n)) -(3*(n*n|^2
    )*2+3*(n*n))+(2*n|^2+n)+(n+1)|^5*42)/42 by NEWTON:6
      .=(n+1)*(3*n|^(4+2)*2+3*n|^(4+1)+(6*n|^(3+2)*2+6*(n|^3*n)) -(3*n|^(2+1
    )*2+3*(n*n))+(2*n|^2+n)+(n+1)|^5*42)/42 by NEWTON:6
      .=(n+1)*(3*n|^6*2+3*n|^5+(6*n|^5*2+6*(n|^3*n)) -(3*n|^3*2+3*n|^2)+(2*n
    |^2+n)+(n+1)|^5*42)/42 by WSIERP_1:1
      .=(n+1)*(6*n|^6+3*n|^5+(12*n|^5+6*n|^(3+1)) -(6*n|^3+3*n|^2)+(2*n|^2+n
    )+(n+1)|^5*42)/42 by NEWTON:6
      .=(n+1)*(6*n|^6+6*n|^4+15*n|^5-6*n|^3-1*n|^2+n+(n|^5+5*n|^4+10*n|^3 +
    10*n|^2+5*n+1)*42)/42 by Lm6
      .=(n+1)*(6*n|^6+57*n|^5+216*n|^4+414*n|^3+419*n|^2+211*n+42)/42
      .=(n+1)*((n+2)*(2*(n+1)+1)*(3*(n+1)|^4+6*(n+1)|^3-3*(n+1)+1))/42 by Lm16
      .=(n+1)*(n+1+1)*(2*(n+1)+1)*(3*(n+1)|^4+6*(n+1)|^3-3*(n+1)+1)/42;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=0 |^6 by A1
    .= 0*(0+1)*(2*0+1)*(3*0|^4+6*0|^3-3*0+1)/42 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
