4159 = (2 * 2079) + 1 ;
hence not 2 divides 4159 by NAT_4:9; :: thesis:
4159 = (3 * 1386) + 1 ;
hence not 3 divides 4159 by NAT_4:9; :: thesis:
4159 = (5 * 831) + 4 ;
hence not 5 divides 4159 by NAT_4:9; :: thesis:
4159 = (7 * 594) + 1 ;
hence not 7 divides 4159 by NAT_4:9; :: thesis:
4159 = (11 * 378) + 1 ;
hence not 11 divides 4159 by NAT_4:9; :: thesis:
4159 = (13 * 319) + 12 ;
hence not 13 divides 4159 by NAT_4:9; :: thesis:
4159 = (17 * 244) + 11 ;
hence not 17 divides 4159 by NAT_4:9; :: thesis:
4159 = (19 * 218) + 17 ;
hence not 19 divides 4159 by NAT_4:9; :: thesis:
4159 = (23 * 180) + 19 ;
hence not 23 divides 4159 by NAT_4:9; :: thesis:
4159 = (29 * 143) + 12 ;
hence not 29 divides 4159 by NAT_4:9; :: thesis:
4159 = (31 * 134) + 5 ;
hence not 31 divides 4159 by NAT_4:9; :: thesis:
4159 = (37 * 112) + 15 ;
hence not 37 divides 4159 by NAT_4:9; :: thesis:
4159 = (41 * 101) + 18 ;
hence not 41 divides 4159 by NAT_4:9; :: thesis:
4159 = (43 * 96) + 31 ;
hence not 43 divides 4159 by NAT_4:9; :: thesis:
4159 = (47 * 88) + 23 ;
hence not 47 divides 4159 by NAT_4:9; :: thesis:
4159 = (53 * 78) + 25 ;
hence not 53 divides 4159 by NAT_4:9; :: thesis:
4159 = (59 * 70) + 29 ;
hence not 59 divides 4159 by NAT_4:9; :: thesis:
4159 = (61 * 68) + 11 ;
hence not 61 divides 4159 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4159 & n is prime holds
not n divides 4159 by XPRIMET1:36;
hence 4159 is prime by NAT_4:14; :: thesis: