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 Th32:
  for m being non zero Nat st 6*m+1 is prime holds 6*m+1 = LP<=6n+1(m)
  proof
    let m be non zero Nat such that
A1: 6*m+1 is prime;
    set x = 6*m+1;
    set A = <=6n+1(m);
A2: x in A;
    x in SetPrimes by A1,NEWTON:def 6;
    then
A3: x in A /\ SetPrimes by A2,XBOOLE_0:def 4;
    for a being ExtReal st a in A /\ SetPrimes holds a <= x
    proof
      let a be ExtReal;
      assume a in A /\ SetPrimes;
      then a in A by XBOOLE_0:def 4;
      hence a <= x by Th7;
    end;
    hence thesis by A3,XXREAL_2:def 8;
  end;
