now :: thesis: ( not 2 divides 401 & not 3 divides 401 & not 5 divides 401 & not 7 divides 401 & not 11 divides 401 & not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )

401 = (2 * 200) + 1 ;
hence not 2 divides 401 by NAT_4:9; :: thesis: ( not 3 divides 401 & not 5 divides 401 & not 7 divides 401 & not 11 divides 401 & not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )
401 = (3 * 133) + 2 ;
hence not 3 divides 401 by NAT_4:9; :: thesis: ( not 5 divides 401 & not 7 divides 401 & not 11 divides 401 & not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )
401 = (5 * 80) + 1 ;
hence not 5 divides 401 by NAT_4:9; :: thesis: ( not 7 divides 401 & not 11 divides 401 & not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )
401 = (7 * 57) + 2 ;
hence not 7 divides 401 by NAT_4:9; :: thesis: ( not 11 divides 401 & not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )
401 = (11 * 36) + 5 ;
hence not 11 divides 401 by NAT_4:9; :: thesis: ( not 13 divides 401 & not 17 divides 401 & not 19 divides 401 )
401 = (13 * 30) + 11 ;
hence not 13 divides 401 by NAT_4:9; :: thesis: ( not 17 divides 401 & not 19 divides 401 )
401 = (17 * 23) + 10 ;
hence not 17 divides 401 by NAT_4:9; :: thesis: not 19 divides 401
401 = (19 * 21) + 2 ;
hence not 19 divides 401 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 401 & n is prime holds
not n divides 401 by XPRIMET1:16;
hence 401 is prime by NAT_4:14; :: thesis: verum