5801 = (2 * 2900) + 1 ;
hence not 2 divides 5801 by NAT_4:9; :: thesis:
5801 = (3 * 1933) + 2 ;
hence not 3 divides 5801 by NAT_4:9; :: thesis:
5801 = (5 * 1160) + 1 ;
hence not 5 divides 5801 by NAT_4:9; :: thesis:
5801 = (7 * 828) + 5 ;
hence not 7 divides 5801 by NAT_4:9; :: thesis:
5801 = (11 * 527) + 4 ;
hence not 11 divides 5801 by NAT_4:9; :: thesis:
5801 = (13 * 446) + 3 ;
hence not 13 divides 5801 by NAT_4:9; :: thesis:
5801 = (17 * 341) + 4 ;
hence not 17 divides 5801 by NAT_4:9; :: thesis:
5801 = (19 * 305) + 6 ;
hence not 19 divides 5801 by NAT_4:9; :: thesis:
5801 = (23 * 252) + 5 ;
hence not 23 divides 5801 by NAT_4:9; :: thesis:
5801 = (29 * 200) + 1 ;
hence not 29 divides 5801 by NAT_4:9; :: thesis:
5801 = (31 * 187) + 4 ;
hence not 31 divides 5801 by NAT_4:9; :: thesis:
5801 = (37 * 156) + 29 ;
hence not 37 divides 5801 by NAT_4:9; :: thesis:
5801 = (41 * 141) + 20 ;
hence not 41 divides 5801 by NAT_4:9; :: thesis:
5801 = (43 * 134) + 39 ;
hence not 43 divides 5801 by NAT_4:9; :: thesis:
5801 = (47 * 123) + 20 ;
hence not 47 divides 5801 by NAT_4:9; :: thesis:
5801 = (53 * 109) + 24 ;
hence not 53 divides 5801 by NAT_4:9; :: thesis:
5801 = (59 * 98) + 19 ;
hence not 59 divides 5801 by NAT_4:9; :: thesis:
5801 = (61 * 95) + 6 ;
hence not 61 divides 5801 by NAT_4:9; :: thesis:
5801 = (67 * 86) + 39 ;
hence not 67 divides 5801 by NAT_4:9; :: thesis:
5801 = (71 * 81) + 50 ;
hence not 71 divides 5801 by NAT_4:9; :: thesis:
5801 = (73 * 79) + 34 ;
hence not 73 divides 5801 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5801 & n is prime holds
not n divides 5801 by XPRIMET1:42;
hence 5801 is prime by NAT_4:14; :: thesis: