now :: thesis: ( not 2 divides 31 & not 3 divides 31 & not 5 divides 31 )

31 = (2 * 15) + 1 ;
hence not 2 divides 31 by NAT_4:9; :: thesis: ( not 3 divides 31 & not 5 divides 31 )
31 = (3 * 10) + 1 ;
hence not 3 divides 31 by NAT_4:9; :: thesis: not 5 divides 31
31 = (5 * 6) + 1 ;
hence not 5 divides 31 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 31 & n is prime holds
not n divides 31 by XPRIMET1:6;
hence 31 is prime by NAT_4:14; :: thesis: verum