now :: thesis: ( not 2 divides 251 & not 3 divides 251 & not 5 divides 251 & not 7 divides 251 & not 11 divides 251 & not 13 divides 251 )

251 = (2 * 125) + 1 ;
hence not 2 divides 251 by NAT_4:9; :: thesis: ( not 3 divides 251 & not 5 divides 251 & not 7 divides 251 & not 11 divides 251 & not 13 divides 251 )
251 = (3 * 83) + 2 ;
hence not 3 divides 251 by NAT_4:9; :: thesis: ( not 5 divides 251 & not 7 divides 251 & not 11 divides 251 & not 13 divides 251 )
251 = (5 * 50) + 1 ;
hence not 5 divides 251 by NAT_4:9; :: thesis: ( not 7 divides 251 & not 11 divides 251 & not 13 divides 251 )
251 = (7 * 35) + 6 ;
hence not 7 divides 251 by NAT_4:9; :: thesis: ( not 11 divides 251 & not 13 divides 251 )
251 = (11 * 22) + 9 ;
hence not 11 divides 251 by NAT_4:9; :: thesis: not 13 divides 251
251 = (13 * 19) + 4 ;
hence not 13 divides 251 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 251 & n is prime holds
not n divides 251 by XPRIMET1:12;
hence 251 is prime by NAT_4:14; :: thesis: verum