now :: thesis: ( not 2 divides 601 & not 3 divides 601 & not 5 divides 601 & not 7 divides 601 & not 11 divides 601 & not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )

601 = (2 * 300) + 1 ;
hence not 2 divides 601 by NAT_4:9; :: thesis: ( not 3 divides 601 & not 5 divides 601 & not 7 divides 601 & not 11 divides 601 & not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (3 * 200) + 1 ;
hence not 3 divides 601 by NAT_4:9; :: thesis: ( not 5 divides 601 & not 7 divides 601 & not 11 divides 601 & not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (5 * 120) + 1 ;
hence not 5 divides 601 by NAT_4:9; :: thesis: ( not 7 divides 601 & not 11 divides 601 & not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (7 * 85) + 6 ;
hence not 7 divides 601 by NAT_4:9; :: thesis: ( not 11 divides 601 & not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (11 * 54) + 7 ;
hence not 11 divides 601 by NAT_4:9; :: thesis: ( not 13 divides 601 & not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (13 * 46) + 3 ;
hence not 13 divides 601 by NAT_4:9; :: thesis: ( not 17 divides 601 & not 19 divides 601 & not 23 divides 601 )
601 = (17 * 35) + 6 ;
hence not 17 divides 601 by NAT_4:9; :: thesis: ( not 19 divides 601 & not 23 divides 601 )
601 = (19 * 31) + 12 ;
hence not 19 divides 601 by NAT_4:9; :: thesis: not 23 divides 601
601 = (23 * 26) + 3 ;
hence not 23 divides 601 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 601 & n is prime holds
not n divides 601 by XPRIMET1:18;
hence 601 is prime by NAT_4:14; :: thesis: verum