now :: thesis: ( not 2 divides 181 & not 3 divides 181 & not 5 divides 181 & not 7 divides 181 & not 11 divides 181 & not 13 divides 181 )

181 = (2 * 90) + 1 ;
hence not 2 divides 181 by NAT_4:9; :: thesis: ( not 3 divides 181 & not 5 divides 181 & not 7 divides 181 & not 11 divides 181 & not 13 divides 181 )
181 = (3 * 60) + 1 ;
hence not 3 divides 181 by NAT_4:9; :: thesis: ( not 5 divides 181 & not 7 divides 181 & not 11 divides 181 & not 13 divides 181 )
181 = (5 * 36) + 1 ;
hence not 5 divides 181 by NAT_4:9; :: thesis: ( not 7 divides 181 & not 11 divides 181 & not 13 divides 181 )
181 = (7 * 25) + 6 ;
hence not 7 divides 181 by NAT_4:9; :: thesis: ( not 11 divides 181 & not 13 divides 181 )
181 = (11 * 16) + 5 ;
hence not 11 divides 181 by NAT_4:9; :: thesis: not 13 divides 181
181 = (13 * 13) + 12 ;
hence not 13 divides 181 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 181 & n is prime holds
not n divides 181 by XPRIMET1:12;
hence 181 is prime by NAT_4:14; :: thesis: verum