3119 = (2 * 1559) + 1 ;
hence not 2 divides 3119 by NAT_4:9; :: thesis:
3119 = (3 * 1039) + 2 ;
hence not 3 divides 3119 by NAT_4:9; :: thesis:
3119 = (5 * 623) + 4 ;
hence not 5 divides 3119 by NAT_4:9; :: thesis:
3119 = (7 * 445) + 4 ;
hence not 7 divides 3119 by NAT_4:9; :: thesis:
3119 = (11 * 283) + 6 ;
hence not 11 divides 3119 by NAT_4:9; :: thesis:
3119 = (13 * 239) + 12 ;
hence not 13 divides 3119 by NAT_4:9; :: thesis:
3119 = (17 * 183) + 8 ;
hence not 17 divides 3119 by NAT_4:9; :: thesis:
3119 = (19 * 164) + 3 ;
hence not 19 divides 3119 by NAT_4:9; :: thesis:
3119 = (23 * 135) + 14 ;
hence not 23 divides 3119 by NAT_4:9; :: thesis:
3119 = (29 * 107) + 16 ;
hence not 29 divides 3119 by NAT_4:9; :: thesis:
3119 = (31 * 100) + 19 ;
hence not 31 divides 3119 by NAT_4:9; :: thesis:
3119 = (37 * 84) + 11 ;
hence not 37 divides 3119 by NAT_4:9; :: thesis:
3119 = (41 * 76) + 3 ;
hence not 41 divides 3119 by NAT_4:9; :: thesis:
3119 = (43 * 72) + 23 ;
hence not 43 divides 3119 by NAT_4:9; :: thesis:
3119 = (47 * 66) + 17 ;
hence not 47 divides 3119 by NAT_4:9; :: thesis:
3119 = (53 * 58) + 45 ;
hence not 53 divides 3119 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3119 & n is prime holds
not n divides 3119 by XPRIMET1:32;
hence 3119 is prime by NAT_4:14; :: thesis: