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 Th44:
  for n being non zero Nat
  for s being Nat st n <= s holds PrimeNumbersFS(s).n = primenumber(n-1)
  proof
    let n be non zero Nat;
    let s be Nat;
    reconsider n1 = n-1 as Element of NAT by INT_1:3;
    assume
A1: n <= s;
    n1 < n-0 by XREAL_1:8;
    then n1 < s by A1,XXREAL_0:2;
    then PrimeNumbersFS(s).(n1+1) = primenumber(n1) by Th43;
    hence thesis;
  end;
