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;

theorem Th50:
  for k being non zero Nat st k <= s holds
  Problem58Solution(s).k = sequenceQk(s).k |^ primenumber(k-1)
  proof
    let k be non zero Nat such that
A1: k <= s;
    set p = primenumber(k-1);
    set AnPk = sequenceAnPk(s,k);
    len AnPk = s by A1,Def10;
    then
    reconsider AnPk as natural-valueds-element FinSequence by CARD_1:def 7;
    set Ak1Pk = sequenceAk1Pk(s);
    set I = idseq(s);
    set A = sequenceA(s);
    set Qk = sequenceQk(s);
    set F = I to_power AnPk;
A2: len F = s by CARD_1:def 7;
    then
A3: dom F = Seg s by FINSEQ_1:def 3;
    1 <= k by NAT_1:14;
    then
A4: k in dom F by A1,A2,FINSEQ_3:25;
    Ak1Pk.k = (A.k + 1) / p by A1,Def9;
    then (Ak1Pk.k) * p = A.k + 1 by XCMPLX_1:87;
    then
A5: k |^ (Ak1Pk.k) |^ p = k|^(A.k+1) by NEWTON:9;
    F is FinSequence of REAL by FINSEQ_1:102;
    then
A6: Product(F) |^ p = Product(F|^p) by NAT_3:15;
A7: len(F|^p) = len F by NAT_3:def 1;
A8: len(I to_power A) = s by CARD_1:def 7;
A9: 1 <= k by NAT_1:14;
    then
A10: k in dom(F|^p) by A1,A2,A7,FINSEQ_3:25;
    then
A11: (F|^p).k = F.k |^ p by NAT_3:def 1;
A12: I.k = k by A3,A4,FINSEQ_2:49;
A13: F.k = (I.k) to_power (AnPk.k) by A4,Def6;
A14: AnPk.k = 0 by A1,Def10;
A15: k |^ 0 = 1 by NEWTON:4;
    len A = s by Def5;
    then
A16: dom A = Seg s by FINSEQ_1:def 3;
    dom (I to_power A) = dom I /\ dom A by Def6;
    then
A17: (I to_power A).k = k to_power (A.k) by A3,A4,A12,A16,Def6
    .= k |^ A.k * (F|^p).k by A12,A11,A13,A14,A15;
    for j being Nat st j in dom(F|^p) & k <> j holds
    (I to_power A).j = (F|^p).j
    proof
      let j be Nat;
      assume j in dom(F|^p);
      then j <> 0 & j <= s by A2,A7,FINSEQ_3:25;
      hence thesis by A1,Th49;
    end;
    then
A18: Product (I to_power A) = k |^ A.k * Product(F) |^ p
    by A6,A7,A8,A10,A17,Th4,CARD_1:def 7;
    thus Problem58Solution(s).k = k * numberQ(s) by A1,A9,Th48
    .= k * k |^ A.k * Product(F) |^ p by A18
    .= k |^ (Ak1Pk.k) |^ p * Product(F) |^ p by A5,NEWTON:6
    .= ((k |^ (Ak1Pk.k)) * Product(F)) |^ p by NEWTON:7
    .= Qk.k |^ p by A1,Def11;
  end;
