reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;

theorem
  <*15,20,12*> is a_solution_of_Sierp168
  proof
    set f = <*15,20,12*>;
    set h = f|2;
    set g = h" ^2;
    g = <*1/(15^2),1/(20^2)*>
    proof
      dom g = dom(h") /\ dom(h") by VALUED_1:def 4;
      then
A1:   len g = len h by VALUED_1:def 7,FINSEQ_3:29;
      len f = 3 by FINSEQ_1:45;
      then
A2:   len h = 2 by FINSEQ_1:59;
      hence len g = len <*1/(15^2),1/(20^2)*> by A1,FINSEQ_1:44;
      let k;
      assume that
      1 <= k and
A3:   k <= len g;
A4:   g.k = (h".k)^2 by VALUED_1:11;
A5:   h".k = (h.k)" by VALUED_1:10;
      k = 0 or ... or k = 2 by A1,A2,A3;
      hence g.k = <*1/(15^2),1/(20^2)*>.k by A4,A5;
    end;
    then Sum g = 1/(15^2) + 1/(20^2) by RVSUM_1:77;
    hence thesis;
  end;
