5869 = (2 * 2934) + 1 ;
hence not 2 divides 5869 by NAT_4:9; :: thesis:
5869 = (3 * 1956) + 1 ;
hence not 3 divides 5869 by NAT_4:9; :: thesis:
5869 = (5 * 1173) + 4 ;
hence not 5 divides 5869 by NAT_4:9; :: thesis:
5869 = (7 * 838) + 3 ;
hence not 7 divides 5869 by NAT_4:9; :: thesis:
5869 = (11 * 533) + 6 ;
hence not 11 divides 5869 by NAT_4:9; :: thesis:
5869 = (13 * 451) + 6 ;
hence not 13 divides 5869 by NAT_4:9; :: thesis:
5869 = (17 * 345) + 4 ;
hence not 17 divides 5869 by NAT_4:9; :: thesis:
5869 = (19 * 308) + 17 ;
hence not 19 divides 5869 by NAT_4:9; :: thesis:
5869 = (23 * 255) + 4 ;
hence not 23 divides 5869 by NAT_4:9; :: thesis:
5869 = (29 * 202) + 11 ;
hence not 29 divides 5869 by NAT_4:9; :: thesis:
5869 = (31 * 189) + 10 ;
hence not 31 divides 5869 by NAT_4:9; :: thesis:
5869 = (37 * 158) + 23 ;
hence not 37 divides 5869 by NAT_4:9; :: thesis:
5869 = (41 * 143) + 6 ;
hence not 41 divides 5869 by NAT_4:9; :: thesis:
5869 = (43 * 136) + 21 ;
hence not 43 divides 5869 by NAT_4:9; :: thesis:
5869 = (47 * 124) + 41 ;
hence not 47 divides 5869 by NAT_4:9; :: thesis:
5869 = (53 * 110) + 39 ;
hence not 53 divides 5869 by NAT_4:9; :: thesis:
5869 = (59 * 99) + 28 ;
hence not 59 divides 5869 by NAT_4:9; :: thesis:
5869 = (61 * 96) + 13 ;
hence not 61 divides 5869 by NAT_4:9; :: thesis:
5869 = (67 * 87) + 40 ;
hence not 67 divides 5869 by NAT_4:9; :: thesis:
5869 = (71 * 82) + 47 ;
hence not 71 divides 5869 by NAT_4:9; :: thesis:
5869 = (73 * 80) + 29 ;
hence not 73 divides 5869 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5869 & n is prime holds
not n divides 5869 by XPRIMET1:42;
hence 5869 is prime by NAT_4:14; :: thesis: