5839 = (2 * 2919) + 1 ;
hence not 2 divides 5839 by NAT_4:9; :: thesis:
5839 = (3 * 1946) + 1 ;
hence not 3 divides 5839 by NAT_4:9; :: thesis:
5839 = (5 * 1167) + 4 ;
hence not 5 divides 5839 by NAT_4:9; :: thesis:
5839 = (7 * 834) + 1 ;
hence not 7 divides 5839 by NAT_4:9; :: thesis:
5839 = (11 * 530) + 9 ;
hence not 11 divides 5839 by NAT_4:9; :: thesis:
5839 = (13 * 449) + 2 ;
hence not 13 divides 5839 by NAT_4:9; :: thesis:
5839 = (17 * 343) + 8 ;
hence not 17 divides 5839 by NAT_4:9; :: thesis:
5839 = (19 * 307) + 6 ;
hence not 19 divides 5839 by NAT_4:9; :: thesis:
5839 = (23 * 253) + 20 ;
hence not 23 divides 5839 by NAT_4:9; :: thesis:
5839 = (29 * 201) + 10 ;
hence not 29 divides 5839 by NAT_4:9; :: thesis:
5839 = (31 * 188) + 11 ;
hence not 31 divides 5839 by NAT_4:9; :: thesis:
5839 = (37 * 157) + 30 ;
hence not 37 divides 5839 by NAT_4:9; :: thesis:
5839 = (41 * 142) + 17 ;
hence not 41 divides 5839 by NAT_4:9; :: thesis:
5839 = (43 * 135) + 34 ;
hence not 43 divides 5839 by NAT_4:9; :: thesis:
5839 = (47 * 124) + 11 ;
hence not 47 divides 5839 by NAT_4:9; :: thesis:
5839 = (53 * 110) + 9 ;
hence not 53 divides 5839 by NAT_4:9; :: thesis:
5839 = (59 * 98) + 57 ;
hence not 59 divides 5839 by NAT_4:9; :: thesis:
5839 = (61 * 95) + 44 ;
hence not 61 divides 5839 by NAT_4:9; :: thesis:
5839 = (67 * 87) + 10 ;
hence not 67 divides 5839 by NAT_4:9; :: thesis:
5839 = (71 * 82) + 17 ;
hence not 71 divides 5839 by NAT_4:9; :: thesis:
5839 = (73 * 79) + 72 ;
hence not 73 divides 5839 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5839 & n is prime holds
not n divides 5839 by XPRIMET1:42;
hence 5839 is prime by NAT_4:14; :: thesis: