3719 = (2 * 1859) + 1 ;
hence not 2 divides 3719 by NAT_4:9; :: thesis:
3719 = (3 * 1239) + 2 ;
hence not 3 divides 3719 by NAT_4:9; :: thesis:
3719 = (5 * 743) + 4 ;
hence not 5 divides 3719 by NAT_4:9; :: thesis:
3719 = (7 * 531) + 2 ;
hence not 7 divides 3719 by NAT_4:9; :: thesis:
3719 = (11 * 338) + 1 ;
hence not 11 divides 3719 by NAT_4:9; :: thesis:
3719 = (13 * 286) + 1 ;
hence not 13 divides 3719 by NAT_4:9; :: thesis:
3719 = (17 * 218) + 13 ;
hence not 17 divides 3719 by NAT_4:9; :: thesis:
3719 = (19 * 195) + 14 ;
hence not 19 divides 3719 by NAT_4:9; :: thesis:
3719 = (23 * 161) + 16 ;
hence not 23 divides 3719 by NAT_4:9; :: thesis:
3719 = (29 * 128) + 7 ;
hence not 29 divides 3719 by NAT_4:9; :: thesis:
3719 = (31 * 119) + 30 ;
hence not 31 divides 3719 by NAT_4:9; :: thesis:
3719 = (37 * 100) + 19 ;
hence not 37 divides 3719 by NAT_4:9; :: thesis:
3719 = (41 * 90) + 29 ;
hence not 41 divides 3719 by NAT_4:9; :: thesis:
3719 = (43 * 86) + 21 ;
hence not 43 divides 3719 by NAT_4:9; :: thesis:
3719 = (47 * 79) + 6 ;
hence not 47 divides 3719 by NAT_4:9; :: thesis:
3719 = (53 * 70) + 9 ;
hence not 53 divides 3719 by NAT_4:9; :: thesis:
3719 = (59 * 63) + 2 ;
hence not 59 divides 3719 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3719 & n is prime holds
not n divides 3719 by XPRIMET1:34;
hence 3719 is prime by NAT_4:14; :: thesis: