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
  LP<=6n+1(1) = 7
  proof
    set A = <=6n+1(1);
    set X = A /\ SetPrimes;
A1: 7 in A;
    7 in SetPrimes by XPRIMES1:7,NEWTON:def 6;
    then
A2: 7 in X by A1,XBOOLE_0:def 4;
    now
      let x be ExtReal;
      assume x in X;
      then x in A by XBOOLE_0:def 4;
      then ex a being Nat st a = x & a <= 6*1+1;
      hence x <= 7;
    end;
    hence thesis by A2,XXREAL_2:def 8;
  end;
