3037 = (2 * 1518) + 1 ;
hence not 2 divides 3037 by NAT_4:9; :: thesis:
3037 = (3 * 1012) + 1 ;
hence not 3 divides 3037 by NAT_4:9; :: thesis:
3037 = (5 * 607) + 2 ;
hence not 5 divides 3037 by NAT_4:9; :: thesis:
3037 = (7 * 433) + 6 ;
hence not 7 divides 3037 by NAT_4:9; :: thesis:
3037 = (11 * 276) + 1 ;
hence not 11 divides 3037 by NAT_4:9; :: thesis:
3037 = (13 * 233) + 8 ;
hence not 13 divides 3037 by NAT_4:9; :: thesis:
3037 = (17 * 178) + 11 ;
hence not 17 divides 3037 by NAT_4:9; :: thesis:
3037 = (19 * 159) + 16 ;
hence not 19 divides 3037 by NAT_4:9; :: thesis:
3037 = (23 * 132) + 1 ;
hence not 23 divides 3037 by NAT_4:9; :: thesis:
3037 = (29 * 104) + 21 ;
hence not 29 divides 3037 by NAT_4:9; :: thesis:
3037 = (31 * 97) + 30 ;
hence not 31 divides 3037 by NAT_4:9; :: thesis:
3037 = (37 * 82) + 3 ;
hence not 37 divides 3037 by NAT_4:9; :: thesis:
3037 = (41 * 74) + 3 ;
hence not 41 divides 3037 by NAT_4:9; :: thesis:
3037 = (43 * 70) + 27 ;
hence not 43 divides 3037 by NAT_4:9; :: thesis:
3037 = (47 * 64) + 29 ;
hence not 47 divides 3037 by NAT_4:9; :: thesis:
3037 = (53 * 57) + 16 ;
hence not 53 divides 3037 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3037 & n is prime holds
not n divides 3037 by XPRIMET1:32;
hence 3037 is prime by NAT_4:14; :: thesis: