now :: thesis: ( not 2 divides 241 & not 3 divides 241 & not 5 divides 241 & not 7 divides 241 & not 11 divides 241 & not 13 divides 241 )

241 = (2 * 120) + 1 ;
hence not 2 divides 241 by NAT_4:9; :: thesis: ( not 3 divides 241 & not 5 divides 241 & not 7 divides 241 & not 11 divides 241 & not 13 divides 241 )
241 = (3 * 80) + 1 ;
hence not 3 divides 241 by NAT_4:9; :: thesis: ( not 5 divides 241 & not 7 divides 241 & not 11 divides 241 & not 13 divides 241 )
241 = (5 * 48) + 1 ;
hence not 5 divides 241 by NAT_4:9; :: thesis: ( not 7 divides 241 & not 11 divides 241 & not 13 divides 241 )
241 = (7 * 34) + 3 ;
hence not 7 divides 241 by NAT_4:9; :: thesis: ( not 11 divides 241 & not 13 divides 241 )
241 = (11 * 21) + 10 ;
hence not 11 divides 241 by NAT_4:9; :: thesis: not 13 divides 241
241 = (13 * 18) + 7 ;
hence not 13 divides 241 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 241 & n is prime holds
not n divides 241 by XPRIMET1:12;
hence 241 is prime by NAT_4:14; :: thesis: verum