now :: thesis: ( not 2 divides 281 & not 3 divides 281 & not 5 divides 281 & not 7 divides 281 & not 11 divides 281 & not 13 divides 281 )

281 = (2 * 140) + 1 ;
hence not 2 divides 281 by NAT_4:9; :: thesis: ( not 3 divides 281 & not 5 divides 281 & not 7 divides 281 & not 11 divides 281 & not 13 divides 281 )
281 = (3 * 93) + 2 ;
hence not 3 divides 281 by NAT_4:9; :: thesis: ( not 5 divides 281 & not 7 divides 281 & not 11 divides 281 & not 13 divides 281 )
281 = (5 * 56) + 1 ;
hence not 5 divides 281 by NAT_4:9; :: thesis: ( not 7 divides 281 & not 11 divides 281 & not 13 divides 281 )
281 = (7 * 40) + 1 ;
hence not 7 divides 281 by NAT_4:9; :: thesis: ( not 11 divides 281 & not 13 divides 281 )
281 = (11 * 25) + 6 ;
hence not 11 divides 281 by NAT_4:9; :: thesis: not 13 divides 281
281 = (13 * 21) + 8 ;
hence not 13 divides 281 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 281 & n is prime holds
not n divides 281 by XPRIMET1:12;
hence 281 is prime by NAT_4:14; :: thesis: verum