5701 = (2 * 2850) + 1 ;
hence not 2 divides 5701 by NAT_4:9; :: thesis:
5701 = (3 * 1900) + 1 ;
hence not 3 divides 5701 by NAT_4:9; :: thesis:
5701 = (5 * 1140) + 1 ;
hence not 5 divides 5701 by NAT_4:9; :: thesis:
5701 = (7 * 814) + 3 ;
hence not 7 divides 5701 by NAT_4:9; :: thesis:
5701 = (11 * 518) + 3 ;
hence not 11 divides 5701 by NAT_4:9; :: thesis:
5701 = (13 * 438) + 7 ;
hence not 13 divides 5701 by NAT_4:9; :: thesis:
5701 = (17 * 335) + 6 ;
hence not 17 divides 5701 by NAT_4:9; :: thesis:
5701 = (19 * 300) + 1 ;
hence not 19 divides 5701 by NAT_4:9; :: thesis:
5701 = (23 * 247) + 20 ;
hence not 23 divides 5701 by NAT_4:9; :: thesis:
5701 = (29 * 196) + 17 ;
hence not 29 divides 5701 by NAT_4:9; :: thesis:
5701 = (31 * 183) + 28 ;
hence not 31 divides 5701 by NAT_4:9; :: thesis:
5701 = (37 * 154) + 3 ;
hence not 37 divides 5701 by NAT_4:9; :: thesis:
5701 = (41 * 139) + 2 ;
hence not 41 divides 5701 by NAT_4:9; :: thesis:
5701 = (43 * 132) + 25 ;
hence not 43 divides 5701 by NAT_4:9; :: thesis:
5701 = (47 * 121) + 14 ;
hence not 47 divides 5701 by NAT_4:9; :: thesis:
5701 = (53 * 107) + 30 ;
hence not 53 divides 5701 by NAT_4:9; :: thesis:
5701 = (59 * 96) + 37 ;
hence not 59 divides 5701 by NAT_4:9; :: thesis:
5701 = (61 * 93) + 28 ;
hence not 61 divides 5701 by NAT_4:9; :: thesis:
5701 = (67 * 85) + 6 ;
hence not 67 divides 5701 by NAT_4:9; :: thesis:
5701 = (71 * 80) + 21 ;
hence not 71 divides 5701 by NAT_4:9; :: thesis:
5701 = (73 * 78) + 7 ;
hence not 73 divides 5701 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5701 & n is prime holds
not n divides 5701 by XPRIMET1:42;
hence 5701 is prime by NAT_4:14; :: thesis: