5843 = (2 * 2921) + 1 ;
hence not 2 divides 5843 by NAT_4:9; :: thesis:
5843 = (3 * 1947) + 2 ;
hence not 3 divides 5843 by NAT_4:9; :: thesis:
5843 = (5 * 1168) + 3 ;
hence not 5 divides 5843 by NAT_4:9; :: thesis:
5843 = (7 * 834) + 5 ;
hence not 7 divides 5843 by NAT_4:9; :: thesis:
5843 = (11 * 531) + 2 ;
hence not 11 divides 5843 by NAT_4:9; :: thesis:
5843 = (13 * 449) + 6 ;
hence not 13 divides 5843 by NAT_4:9; :: thesis:
5843 = (17 * 343) + 12 ;
hence not 17 divides 5843 by NAT_4:9; :: thesis:
5843 = (19 * 307) + 10 ;
hence not 19 divides 5843 by NAT_4:9; :: thesis:
5843 = (23 * 254) + 1 ;
hence not 23 divides 5843 by NAT_4:9; :: thesis:
5843 = (29 * 201) + 14 ;
hence not 29 divides 5843 by NAT_4:9; :: thesis:
5843 = (31 * 188) + 15 ;
hence not 31 divides 5843 by NAT_4:9; :: thesis:
5843 = (37 * 157) + 34 ;
hence not 37 divides 5843 by NAT_4:9; :: thesis:
5843 = (41 * 142) + 21 ;
hence not 41 divides 5843 by NAT_4:9; :: thesis:
5843 = (43 * 135) + 38 ;
hence not 43 divides 5843 by NAT_4:9; :: thesis:
5843 = (47 * 124) + 15 ;
hence not 47 divides 5843 by NAT_4:9; :: thesis:
5843 = (53 * 110) + 13 ;
hence not 53 divides 5843 by NAT_4:9; :: thesis:
5843 = (59 * 99) + 2 ;
hence not 59 divides 5843 by NAT_4:9; :: thesis:
5843 = (61 * 95) + 48 ;
hence not 61 divides 5843 by NAT_4:9; :: thesis:
5843 = (67 * 87) + 14 ;
hence not 67 divides 5843 by NAT_4:9; :: thesis:
5843 = (71 * 82) + 21 ;
hence not 71 divides 5843 by NAT_4:9; :: thesis:
5843 = (73 * 80) + 3 ;
hence not 73 divides 5843 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5843 & n is prime holds
not n divides 5843 by XPRIMET1:42;
hence 5843 is prime by NAT_4:14; :: thesis: