5857 = (2 * 2928) + 1 ;
hence not 2 divides 5857 by NAT_4:9; :: thesis:
5857 = (3 * 1952) + 1 ;
hence not 3 divides 5857 by NAT_4:9; :: thesis:
5857 = (5 * 1171) + 2 ;
hence not 5 divides 5857 by NAT_4:9; :: thesis:
5857 = (7 * 836) + 5 ;
hence not 7 divides 5857 by NAT_4:9; :: thesis:
5857 = (11 * 532) + 5 ;
hence not 11 divides 5857 by NAT_4:9; :: thesis:
5857 = (13 * 450) + 7 ;
hence not 13 divides 5857 by NAT_4:9; :: thesis:
5857 = (17 * 344) + 9 ;
hence not 17 divides 5857 by NAT_4:9; :: thesis:
5857 = (19 * 308) + 5 ;
hence not 19 divides 5857 by NAT_4:9; :: thesis:
5857 = (23 * 254) + 15 ;
hence not 23 divides 5857 by NAT_4:9; :: thesis:
5857 = (29 * 201) + 28 ;
hence not 29 divides 5857 by NAT_4:9; :: thesis:
5857 = (31 * 188) + 29 ;
hence not 31 divides 5857 by NAT_4:9; :: thesis:
5857 = (37 * 158) + 11 ;
hence not 37 divides 5857 by NAT_4:9; :: thesis:
5857 = (41 * 142) + 35 ;
hence not 41 divides 5857 by NAT_4:9; :: thesis:
5857 = (43 * 136) + 9 ;
hence not 43 divides 5857 by NAT_4:9; :: thesis:
5857 = (47 * 124) + 29 ;
hence not 47 divides 5857 by NAT_4:9; :: thesis:
5857 = (53 * 110) + 27 ;
hence not 53 divides 5857 by NAT_4:9; :: thesis:
5857 = (59 * 99) + 16 ;
hence not 59 divides 5857 by NAT_4:9; :: thesis:
5857 = (61 * 96) + 1 ;
hence not 61 divides 5857 by NAT_4:9; :: thesis:
5857 = (67 * 87) + 28 ;
hence not 67 divides 5857 by NAT_4:9; :: thesis:
5857 = (71 * 82) + 35 ;
hence not 71 divides 5857 by NAT_4:9; :: thesis:
5857 = (73 * 80) + 17 ;
hence not 73 divides 5857 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5857 & n is prime holds
not n divides 5857 by XPRIMET1:42;
hence 5857 is prime by NAT_4:14; :: thesis: