5153 = (2 * 2576) + 1 ;
hence not 2 divides 5153 by NAT_4:9; :: thesis:
5153 = (3 * 1717) + 2 ;
hence not 3 divides 5153 by NAT_4:9; :: thesis:
5153 = (5 * 1030) + 3 ;
hence not 5 divides 5153 by NAT_4:9; :: thesis:
5153 = (7 * 736) + 1 ;
hence not 7 divides 5153 by NAT_4:9; :: thesis:
5153 = (11 * 468) + 5 ;
hence not 11 divides 5153 by NAT_4:9; :: thesis:
5153 = (13 * 396) + 5 ;
hence not 13 divides 5153 by NAT_4:9; :: thesis:
5153 = (17 * 303) + 2 ;
hence not 17 divides 5153 by NAT_4:9; :: thesis:
5153 = (19 * 271) + 4 ;
hence not 19 divides 5153 by NAT_4:9; :: thesis:
5153 = (23 * 224) + 1 ;
hence not 23 divides 5153 by NAT_4:9; :: thesis:
5153 = (29 * 177) + 20 ;
hence not 29 divides 5153 by NAT_4:9; :: thesis:
5153 = (31 * 166) + 7 ;
hence not 31 divides 5153 by NAT_4:9; :: thesis:
5153 = (37 * 139) + 10 ;
hence not 37 divides 5153 by NAT_4:9; :: thesis:
5153 = (41 * 125) + 28 ;
hence not 41 divides 5153 by NAT_4:9; :: thesis:
5153 = (43 * 119) + 36 ;
hence not 43 divides 5153 by NAT_4:9; :: thesis:
5153 = (47 * 109) + 30 ;
hence not 47 divides 5153 by NAT_4:9; :: thesis:
5153 = (53 * 97) + 12 ;
hence not 53 divides 5153 by NAT_4:9; :: thesis:
5153 = (59 * 87) + 20 ;
hence not 59 divides 5153 by NAT_4:9; :: thesis:
5153 = (61 * 84) + 29 ;
hence not 61 divides 5153 by NAT_4:9; :: thesis:
5153 = (67 * 76) + 61 ;
hence not 67 divides 5153 by NAT_4:9; :: thesis:
5153 = (71 * 72) + 41 ;
hence not 71 divides 5153 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5153 & n is prime holds
not n divides 5153 by XPRIMET1:40;
hence 5153 is prime by NAT_4:14; :: thesis: