now :: thesis: ( not 2 divides 631 & not 3 divides 631 & not 5 divides 631 & not 7 divides 631 & not 11 divides 631 & not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )

631 = (2 * 315) + 1 ;
hence not 2 divides 631 by NAT_4:9; :: thesis: ( not 3 divides 631 & not 5 divides 631 & not 7 divides 631 & not 11 divides 631 & not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (3 * 210) + 1 ;
hence not 3 divides 631 by NAT_4:9; :: thesis: ( not 5 divides 631 & not 7 divides 631 & not 11 divides 631 & not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (5 * 126) + 1 ;
hence not 5 divides 631 by NAT_4:9; :: thesis: ( not 7 divides 631 & not 11 divides 631 & not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (7 * 90) + 1 ;
hence not 7 divides 631 by NAT_4:9; :: thesis: ( not 11 divides 631 & not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (11 * 57) + 4 ;
hence not 11 divides 631 by NAT_4:9; :: thesis: ( not 13 divides 631 & not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (13 * 48) + 7 ;
hence not 13 divides 631 by NAT_4:9; :: thesis: ( not 17 divides 631 & not 19 divides 631 & not 23 divides 631 )
631 = (17 * 37) + 2 ;
hence not 17 divides 631 by NAT_4:9; :: thesis: ( not 19 divides 631 & not 23 divides 631 )
631 = (19 * 33) + 4 ;
hence not 19 divides 631 by NAT_4:9; :: thesis: not 23 divides 631
631 = (23 * 27) + 10 ;
hence not 23 divides 631 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 631 & n is prime holds
not n divides 631 by XPRIMET1:18;
hence 631 is prime by NAT_4:14; :: thesis: verum