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 Th47:
  for k, n being non zero Nat st k <> n & n <= s & k <= s
  holds primenumber(k-1) divides sequenceA(s).n
  proof
    let k, n be non zero Nat;
    assume that
A1: k <> n and
A2: n <= s and
A3: k <= s;
    reconsider s as non zero Nat by A2;
    set A = sequenceA(s);
    set p = primenumber(k-1);
    set r = primenumber(n-1);
    set g = PrimeNumbersFS(s);
A4: 0+1 <= k by NAT_1:13;
A5: len g = s by Th42;
    then
A6: k in dom g by A3,A4,FINSEQ_3:25;
A7: g.k = p by A3,Th44;
    n-1 < s-0 by A2,XREAL_1:8;
    then reconsider e = Product g / r as Nat by Th45;
    set y = CRT(0,e,-1,r);
    1 <= n by NAT_1:14;
    then
A8: n in dom g by A5,A2,FINSEQ_3:25;
A9: A.n = y + Product g by A2,Def5;
A10: g.n = r by A2,Th44;
    then r, e are_coprime by A8,INT_6:25;
    then y solves_CRT 0,e,-1,r by NUMBER14:def 2;
    then
A11: y,0 are_congruent_mod e;
    e divides Product g by NUMBER14:3;
    then
A12: e divides y + Product g by A11,NAT_D:8;
    consider z being Integer such that
A13: z * g.k = Product(g) / g.n by A1,A6,INT_6:12;
    g.k divides z*g.k;
    hence thesis by A7,A9,A10,A12,A13,INT_2:9;
  end;
