
theorem
  127 is prime
proof
  now
    127 = 2*63 + 1; hence not 2 divides 127 by NAT_4:9;
    127 = 3*42 + 1; hence not 3 divides 127 by NAT_4:9;
    127 = 5*25 + 2; hence not 5 divides 127 by NAT_4:9;
    127 = 7*18 + 1; hence not 7 divides 127 by NAT_4:9;
    127 = 11*11 + 6; hence not 11 divides 127 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 127 & n is prime
  holds not n divides 127 by XPRIMET1:10;
  hence thesis by NAT_4:14;
