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 Th49:
  for k, w being non zero Nat st k <= s & w <= s & w <> k holds
  (idseq(s) to_power sequenceA(s)).w =
  ((idseq(s) to_power sequenceAnPk(s,k)) |^ primenumber(k-1)).w
  proof
    let k, w be non zero Nat such that
A1: k <= s and
A2: w <= s and
A3: w <> k;
A4: 1 <= w by NAT_1:14;
    set p = primenumber(k-1);
    set AnPk = sequenceAnPk(s,k);
    len AnPk = s by A1,Def10;
    then
    reconsider AnPk as natural-valued s-element FinSequence by CARD_1:def 7;
    set I = idseq(s);
    set A = sequenceA(s);
    set f = I to_power A;
    set F = I to_power AnPk;
A5: len (F|^p) = len F by NAT_3:def 1;
A6: len F = s by CARD_1:def 7;
    then
A7: w in dom(F|^p) by A2,A4,A5,FINSEQ_3:25;
A8: w in Seg(s) by A2,A4;
    then
A9: I.w = w by FINSEQ_2:49;
    dom F = Seg s by A6,FINSEQ_1:def 3;
    then
A10: F.w = (I.w) to_power (AnPk.w) by A8,Def6;
A11: AnPk.w = A.w / p by A1,A2,A3,Def10;
     len f = s by CARD_1:def 7;
     then w in dom f by A2,A4,FINSEQ_3:25;
     then f.w = (I.w) to_power (A.w) by Def6
     .= w |^ (AnPk.w * p) by A9,A11,XCMPLX_1:87
     .= F.w |^ p by A9,A10,NEWTON:9
     .= (F|^p).w by A7,NAT_3:def 1;
     hence thesis;
   end;
