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 Th70:
  for s being positive Nat
  ex f being (s+1)-element non-empty natural-valued FinSequence st
  f is a_solution_of_Sierp168
  proof
    defpred P[Nat] means
    ex f being ($1+1)-element non-empty natural-valued FinSequence st
    f is a_solution_of_Sierp168;
A1: P[1] by Th67;
A2: for s being non zero Nat st P[s] holds P[s+1]
    proof
      let s be non zero Nat;
      given T being (s+1)-element non-empty natural-valued FinSequence
      such that
A3:   T is a_solution_of_Sierp168;
      take X = SierpProblem168FS(T);
A4:   12(#)T|(s-1) = (12(#)T)|(s-1) by Th56;
A5:   len <*15*T.s,20*T.s*> = 2 by FINSEQ_1:44;
A6:   len (12(#)T) = len T by COMPLSP2:3;
A7:   len T = s+1 by CARD_1:def 7;
      then
A8:   len (12(#)T|(s-1)) = s-1 by A4,A6,FINSEQ_1:59,XREAL_1:6;
A9:   len (12(#)T|(s-1)^<*15*T.s,20*T.s*>)
      = len (12(#)T|(s-1)) + len <*15*T.s,20*T.s*> by FINSEQ_1:22
      .= s+1 by A5,A8;
A10:  len <* 15*T.s, 20*T.s, 12*T.(s+1) *> = 3 by FINSEQ_1:45;
      s-1+1+0 <= s+2 & s+2 <= s-1+3 by XREAL_1:6;
      then
A11:  (12(#)T|(s-1)^<* 15*T.s, 20*T.s, 12*T.(s+1) *>).(s+2)
      = <* 15*T.s, 20*T.s, 12*T.(s+1) *>.(s+2-(s-1))
      by A8,A10,FINSEQ_1:23;
A12:  (12"(#)T")^2 = (12"^2)(#)(T"^2) by BORSUK_7:18;
A13:  (T"|s) ^2 = (T"^2)|s by Th62;
A14:  (T|s)" ^2 = (T"|s) ^2 by Th61;
      (T"^2).s = (T".s)^2 by VALUED_1:11;
      then
A15:  (T"^2).s = (T.s)"^2 by VALUED_1:10;
A16:  0+1 <= s by NAT_1:13;
      s <= s+1 by NAT_1:11;
      then s in dom T by A7,A16,FINSEQ_3:25;
      then
A17:  T.s <> 0;
      then
A18:  1/(15*T.s)^2 + 1/(20*T.s)^2
      = (1*(20^2*(T.s)^2)+1*(15^2*(T.s)^2)) / ((15^2*(T.s)^2)*(20^2*(T.s)^2))
      by XCMPLX_1:116
      .= (625*(T.s)^2) / (225*(T.s)^2*400*(T.s)^2)
      .= (1*625) / (12^2*(T.s)^2*625) by A17,XCMPLX_1:91
      .= 1/(12*T.s)^2 by XCMPLX_1:91;
      ((12(#)T)|(s-1))" = (12(#)T)"|(s-1) by Th61;
      then
A19:  Sum (((12(#)T)|(s-1))"^2) = Sum ((12(#)T)"^2|(s-1)) by Th62;
      now
        let c;
        thus (c*T.s)"^2 = (1/(c*T.s))^2
        .= (1*1)/(c*T.s)^2 by XCMPLX_1:76;
      end;
      then
A20:  (15*T.s)"^2 = 1/(15*T.s)^2 & (20*T.s)"^2 = 1/(20*T.s)^2;
      <* 15*T.s, 20*T.s *>"^2 = <* (15*T.s)", (20*T.s)" *>^2 by Th64
      .= <* 1/(15*T.s)^2, 1/(20*T.s)^2 *> by A20,TOPREAL6:11;
      then
A21:  Sum (<* 15*T.s, 20*T.s *>"^2) = 1/(15*T.s)^2 + 1/(20*T.s)^2
      by RVSUM_1:77;
      (12(#)T)" = 12"(#)T" by Th58;
      then
A22:  Sum <*(12(#)T)"^2.s*> = 1/(12^2)*((1/(T.s))^2) by A12,A15,VALUED_1:6
      .= 1/(12^2)*((1^2/(T.s)^2)) by XCMPLX_1:76
      .= (1*1)/((12^2)*(T.s)^2) by XCMPLX_1:76
      .= 1/(12*T.s)^2;
      set p = (12(#)T)"^2;
      dom ((12(#)T)"^2) = dom ((12(#)T)") by VALUED_1:11;
      then
A23:  len ((12(#)T)"^2) = len ((12(#)T)") by FINSEQ_3:29;
      dom ((12(#)T)") = dom (12(#)T) by VALUED_1:def 7;
      then len ((12(#)T)") = len (12(#)T) by FINSEQ_3:29;
      then
A24:  p|(s-1+1) = p|(s-1) ^ <*p.(s-1+1)*> by A6,A7,A23,FINSEQ_5:83,XREAL_1:6;
A25:  (12"^2(#)T"^2)|s = 12"^2*(T"^2|s) by Th56;
A26:  (12"(#)T")^2|s = (12"^2(#)T"^2)|s by BORSUK_7:18;
      12(#)T|(s-1)^(<*15*T.s,20*T.s*>^<*12*T.(s+1)*>) =
      12(#)T|(s-1)^<*15*T.s,20*T.s*>^<*12*T.(s+1)*> by FINSEQ_1:32;
      then (X|(s+1))"
      = (12(#)T|(s-1))" ^ <* 15*T.s, 20*T.s *>" by A9,Th60;
      then (X|(s+1))"^2 = (12(#)T|(s-1))"^2 ^ <* 15*T.s, 20*T.s *>"^2 by Th59;
      hence Sum ((X|(s+1))" ^2)
      = Sum ((12(#)T|(s-1))"^2) + 1/(12*T.s)^2 by A18,A21,RVSUM_1:75
      .= Sum ((12(#)T)"^2|(s-1)) + 1/(12*T.s)^2 by A19,Th56
      .= Sum ((12(#)T)"^2|(s-1) ^ <*(12(#)T)"^2.s*>) by A22,RVSUM_1:74
      .= Sum ((12"(#)T")^2|s) by A24,Th58
      .= (1^2)/(12^2) * (1/(T.(s+1))^2) by A3,A13,A14,A25,A26,RVSUM_2:38
      .= (1*1)/(12^2*(T.(s+1))^2) by XCMPLX_1:76
      .= 1/(X.(s+1+1))^2 by A11;
    end;
    for k being non zero Nat holds P[k] from NAT_1:sch 10(A1,A2);
    hence thesis;
  end;
