2531 = (2 * 1265) + 1 ;
hence not 2 divides 2531 by NAT_4:9; :: thesis:
2531 = (3 * 843) + 2 ;
hence not 3 divides 2531 by NAT_4:9; :: thesis:
2531 = (5 * 506) + 1 ;
hence not 5 divides 2531 by NAT_4:9; :: thesis:
2531 = (7 * 361) + 4 ;
hence not 7 divides 2531 by NAT_4:9; :: thesis:
2531 = (11 * 230) + 1 ;
hence not 11 divides 2531 by NAT_4:9; :: thesis:
2531 = (13 * 194) + 9 ;
hence not 13 divides 2531 by NAT_4:9; :: thesis:
2531 = (17 * 148) + 15 ;
hence not 17 divides 2531 by NAT_4:9; :: thesis:
2531 = (19 * 133) + 4 ;
hence not 19 divides 2531 by NAT_4:9; :: thesis:
2531 = (23 * 110) + 1 ;
hence not 23 divides 2531 by NAT_4:9; :: thesis:
2531 = (29 * 87) + 8 ;
hence not 29 divides 2531 by NAT_4:9; :: thesis:
2531 = (31 * 81) + 20 ;
hence not 31 divides 2531 by NAT_4:9; :: thesis:
2531 = (37 * 68) + 15 ;
hence not 37 divides 2531 by NAT_4:9; :: thesis:
2531 = (41 * 61) + 30 ;
hence not 41 divides 2531 by NAT_4:9; :: thesis:
2531 = (43 * 58) + 37 ;
hence not 43 divides 2531 by NAT_4:9; :: thesis:
2531 = (47 * 53) + 40 ;
hence not 47 divides 2531 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2531 & n is prime holds
not n divides 2531 by XPRIMET1:30;
hence 2531 is prime by NAT_4:14; :: thesis: