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 Th43:
  for n,s being Nat st n < s holds PrimeNumbersFS(s).(n+1) = primenumber(n)
  proof
    let n,s be Nat;
    set h = PrimeNumbersS(s);
    set q = h | s;
    assume
A1: n < s;
    then
A2: n in Segm s by NAT_1:44;
    n+1 <= len q by A1,NAT_1:13;
    hence PrimeNumbersFS(s).(n+1) = q.(n+1-1) by NAT_1:11,AFINSQ_1:def 9
    .= h.n by A2,FUNCT_1:49
    .= primenumber(n) by Def3;
  end;
