now :: thesis: ( not 2 divides 997 & not 3 divides 997 & not 5 divides 997 & not 7 divides 997 & not 11 divides 997 & not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )

997 = (2 * 498) + 1 ;
hence not 2 divides 997 by NAT_4:9; :: thesis: ( not 3 divides 997 & not 5 divides 997 & not 7 divides 997 & not 11 divides 997 & not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (3 * 332) + 1 ;
hence not 3 divides 997 by NAT_4:9; :: thesis: ( not 5 divides 997 & not 7 divides 997 & not 11 divides 997 & not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (5 * 199) + 2 ;
hence not 5 divides 997 by NAT_4:9; :: thesis: ( not 7 divides 997 & not 11 divides 997 & not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (7 * 142) + 3 ;
hence not 7 divides 997 by NAT_4:9; :: thesis: ( not 11 divides 997 & not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (11 * 90) + 7 ;
hence not 11 divides 997 by NAT_4:9; :: thesis: ( not 13 divides 997 & not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (13 * 76) + 9 ;
hence not 13 divides 997 by NAT_4:9; :: thesis: ( not 17 divides 997 & not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (17 * 58) + 11 ;
hence not 17 divides 997 by NAT_4:9; :: thesis: ( not 19 divides 997 & not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (19 * 52) + 9 ;
hence not 19 divides 997 by NAT_4:9; :: thesis: ( not 23 divides 997 & not 29 divides 997 & not 31 divides 997 )
997 = (23 * 43) + 8 ;
hence not 23 divides 997 by NAT_4:9; :: thesis: ( not 29 divides 997 & not 31 divides 997 )
997 = (29 * 34) + 11 ;
hence not 29 divides 997 by NAT_4:9; :: thesis: not 31 divides 997
997 = (31 * 32) + 5 ;
hence not 31 divides 997 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 997 & n is prime holds
not n divides 997 by XPRIMET1:22;
hence 997 is prime by NAT_4:14; :: thesis: verum