5171 = (2 * 2585) + 1 ;
hence not 2 divides 5171 by NAT_4:9; :: thesis:
5171 = (3 * 1723) + 2 ;
hence not 3 divides 5171 by NAT_4:9; :: thesis:
5171 = (5 * 1034) + 1 ;
hence not 5 divides 5171 by NAT_4:9; :: thesis:
5171 = (7 * 738) + 5 ;
hence not 7 divides 5171 by NAT_4:9; :: thesis:
5171 = (11 * 470) + 1 ;
hence not 11 divides 5171 by NAT_4:9; :: thesis:
5171 = (13 * 397) + 10 ;
hence not 13 divides 5171 by NAT_4:9; :: thesis:
5171 = (17 * 304) + 3 ;
hence not 17 divides 5171 by NAT_4:9; :: thesis:
5171 = (19 * 272) + 3 ;
hence not 19 divides 5171 by NAT_4:9; :: thesis:
5171 = (23 * 224) + 19 ;
hence not 23 divides 5171 by NAT_4:9; :: thesis:
5171 = (29 * 178) + 9 ;
hence not 29 divides 5171 by NAT_4:9; :: thesis:
5171 = (31 * 166) + 25 ;
hence not 31 divides 5171 by NAT_4:9; :: thesis:
5171 = (37 * 139) + 28 ;
hence not 37 divides 5171 by NAT_4:9; :: thesis:
5171 = (41 * 126) + 5 ;
hence not 41 divides 5171 by NAT_4:9; :: thesis:
5171 = (43 * 120) + 11 ;
hence not 43 divides 5171 by NAT_4:9; :: thesis:
5171 = (47 * 110) + 1 ;
hence not 47 divides 5171 by NAT_4:9; :: thesis:
5171 = (53 * 97) + 30 ;
hence not 53 divides 5171 by NAT_4:9; :: thesis:
5171 = (59 * 87) + 38 ;
hence not 59 divides 5171 by NAT_4:9; :: thesis:
5171 = (61 * 84) + 47 ;
hence not 61 divides 5171 by NAT_4:9; :: thesis:
5171 = (67 * 77) + 12 ;
hence not 67 divides 5171 by NAT_4:9; :: thesis:
5171 = (71 * 72) + 59 ;
hence not 71 divides 5171 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5171 & n is prime holds
not n divides 5171 by XPRIMET1:40;
hence 5171 is prime by NAT_4:14; :: thesis: