7159 = (2 * 3579) + 1 ;
hence not 2 divides 7159 by NAT_4:9; :: thesis:
7159 = (3 * 2386) + 1 ;
hence not 3 divides 7159 by NAT_4:9; :: thesis:
7159 = (5 * 1431) + 4 ;
hence not 5 divides 7159 by NAT_4:9; :: thesis:
7159 = (7 * 1022) + 5 ;
hence not 7 divides 7159 by NAT_4:9; :: thesis:
7159 = (11 * 650) + 9 ;
hence not 11 divides 7159 by NAT_4:9; :: thesis:
7159 = (13 * 550) + 9 ;
hence not 13 divides 7159 by NAT_4:9; :: thesis:
7159 = (17 * 421) + 2 ;
hence not 17 divides 7159 by NAT_4:9; :: thesis:
7159 = (19 * 376) + 15 ;
hence not 19 divides 7159 by NAT_4:9; :: thesis:
7159 = (23 * 311) + 6 ;
hence not 23 divides 7159 by NAT_4:9; :: thesis:
7159 = (29 * 246) + 25 ;
hence not 29 divides 7159 by NAT_4:9; :: thesis:
7159 = (31 * 230) + 29 ;
hence not 31 divides 7159 by NAT_4:9; :: thesis:
7159 = (37 * 193) + 18 ;
hence not 37 divides 7159 by NAT_4:9; :: thesis:
7159 = (41 * 174) + 25 ;
hence not 41 divides 7159 by NAT_4:9; :: thesis:
7159 = (43 * 166) + 21 ;
hence not 43 divides 7159 by NAT_4:9; :: thesis:
7159 = (47 * 152) + 15 ;
hence not 47 divides 7159 by NAT_4:9; :: thesis:
7159 = (53 * 135) + 4 ;
hence not 53 divides 7159 by NAT_4:9; :: thesis:
7159 = (59 * 121) + 20 ;
hence not 59 divides 7159 by NAT_4:9; :: thesis:
7159 = (61 * 117) + 22 ;
hence not 61 divides 7159 by NAT_4:9; :: thesis:
7159 = (67 * 106) + 57 ;
hence not 67 divides 7159 by NAT_4:9; :: thesis:
7159 = (71 * 100) + 59 ;
hence not 71 divides 7159 by NAT_4:9; :: thesis:
7159 = (73 * 98) + 5 ;
hence not 73 divides 7159 by NAT_4:9; :: thesis:
7159 = (79 * 90) + 49 ;
hence not 79 divides 7159 by NAT_4:9; :: thesis:
7159 = (83 * 86) + 21 ;
hence not 83 divides 7159 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7159 & n is prime holds
not n divides 7159 by XPRIMET1:46;
hence 7159 is prime by NAT_4:14; :: thesis: