 reserve i,j,k,k0,m,n,N for Nat;
 reserve x,y for Real;
 reserve p for Prime;
 reserve s for Real_Sequence;

theorem LtPrimenumber:
  p < primenumber (n+1) implies p <= primenumber n
  proof
    assume A1: p < primenumber (n+1);
    set q1 = primenumber n, q2 = primenumber (n+1);
    card SetPrimenumber p + 1 <= card SetPrimenumber q2
      by A1,CardSetPrime1; then
    card SetPrimenumber p + 1 <= n + 1 by NEWTON:def 8; then
A2: card SetPrimenumber p <= n by XREAL_1:6;
    assume q1 < p; then
    card SetPrimenumber q1 + 1 <= card SetPrimenumber p by CardSetPrime1; then
    n + 1 <= card SetPrimenumber p by NEWTON:def 8; then
    n + 1 <= n by A2,XXREAL_0:2;
    hence thesis by XREAL_1:29;
  end;
