reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th33:
  for m,n being non zero Nat holds
  6*n+1 is prime & m < n implies LP<=6n+1(m) < LP<=6n+1(n)
  proof
    let m,n be non zero Nat;
    set A = <=6n+1(m);
    set x = LP<=6n+1(m);
    set y = LP<=6n+1(n);
    assume that
A1: 6*n+1 is prime and
A2: m < n;
A3: x <= y by A2,Th16;
    now
      assume
A4:   x = y;
      x in A /\ SetPrimes by XXREAL_2:def 8;
      then
A5:   x in A by XBOOLE_0:def 4;
      6*n+1 = y by A1,Th32;
      then 6*n <= 6*m by A4,A5,Th7,XREAL_1:6;
      hence contradiction by A2,XREAL_1:68;
    end;
    hence thesis by A3,XXREAL_0:1;
  end;
