reserve a,b,d,n,k,i,j,x,s for Nat;

theorem Th33:
  for n,m being Nat holds
    n < m iff Product primesFinS n < Product primesFinS m
proof
  let n,m be Nat;
  thus n < m implies Product primesFinS n < Product primesFinS m
  proof
    assume n < m;
    then n+1 <=m by NAT_1:13;
    then
A1:   Product primesFinS (n+1) <= Product primesFinS m by Lm2;
    2 <= primenumber n by MOEBIUS2:8,21;
    then
A2:   Product primesFinS n * 2 <= Product primesFinS n * (primenumber n)
    by XREAL_1:66;
    Product primesFinS n * 1 < Product primesFinS n * 2 by XREAL_1:68;
    then Product primesFinS n < Product primesFinS n * (primenumber n)
    by A2,XXREAL_0:2;
    then Product primesFinS n < Product primesFinS (n+1) by Th25;
    hence thesis by A1,XXREAL_0:2;
  end;
  thus thesis by Lm2;
end;
