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(20) = LP<=6n+1(19)
  proof
    set a = 20;
    set b = 19;
    set X = <=6n+1(a);
    set A = LP<=6n+1(a);
    set Y = <=6n+1(b);
    set B = LP<=6n+1(b);
A1: B in SetPrimes by NEWTON:def 6;
    B in Y /\ SetPrimes by XXREAL_2:def 8;
    then B in Y by XBOOLE_0:def 4;
    then B <= 6*b+1 by Th7;
    then B <= 6*a+1 by XXREAL_0:2;
    then B in X;
    then
A2: B in X /\ SetPrimes by A1,XBOOLE_0:def 4;
    for x being ExtReal st x in X /\ SetPrimes holds x <= B
    proof
      let x be ExtReal such that
A3:   x in X /\ SetPrimes;
A4:   x in SetPrimes by A3,XBOOLE_0:def 4;
      reconsider x as Nat by A3;
      now
        assume x > 115;
        then
A5:     x >= 115+1 by NAT_1:13;
        x in X by A3,XBOOLE_0:def 4;
        then x <= 6*a+1 by Th7;
        then x = 116+0 or ... or x = 116+5 by A5,NAT_1:62;
        hence contradiction
        by A4,NEWTON:def 6,XPRIMES0:116,117,118,119,120,121;
      end;
      then x in Y;
      then x in Y /\ SetPrimes by A4,XBOOLE_0:def 4;
      hence thesis by XXREAL_2:def 8;
    end;
    hence thesis by A2,XXREAL_2:def 8;
  end;
