8419 = (2 * 4209) + 1 ;
hence not 2 divides 8419 by NAT_4:9; :: thesis:
8419 = (3 * 2806) + 1 ;
hence not 3 divides 8419 by NAT_4:9; :: thesis:
8419 = (5 * 1683) + 4 ;
hence not 5 divides 8419 by NAT_4:9; :: thesis:
8419 = (7 * 1202) + 5 ;
hence not 7 divides 8419 by NAT_4:9; :: thesis:
8419 = (11 * 765) + 4 ;
hence not 11 divides 8419 by NAT_4:9; :: thesis:
8419 = (13 * 647) + 8 ;
hence not 13 divides 8419 by NAT_4:9; :: thesis:
8419 = (17 * 495) + 4 ;
hence not 17 divides 8419 by NAT_4:9; :: thesis:
8419 = (19 * 443) + 2 ;
hence not 19 divides 8419 by NAT_4:9; :: thesis:
8419 = (23 * 366) + 1 ;
hence not 23 divides 8419 by NAT_4:9; :: thesis:
8419 = (29 * 290) + 9 ;
hence not 29 divides 8419 by NAT_4:9; :: thesis:
8419 = (31 * 271) + 18 ;
hence not 31 divides 8419 by NAT_4:9; :: thesis:
8419 = (37 * 227) + 20 ;
hence not 37 divides 8419 by NAT_4:9; :: thesis:
8419 = (41 * 205) + 14 ;
hence not 41 divides 8419 by NAT_4:9; :: thesis:
8419 = (43 * 195) + 34 ;
hence not 43 divides 8419 by NAT_4:9; :: thesis:
8419 = (47 * 179) + 6 ;
hence not 47 divides 8419 by NAT_4:9; :: thesis:
8419 = (53 * 158) + 45 ;
hence not 53 divides 8419 by NAT_4:9; :: thesis:
8419 = (59 * 142) + 41 ;
hence not 59 divides 8419 by NAT_4:9; :: thesis:
8419 = (61 * 138) + 1 ;
hence not 61 divides 8419 by NAT_4:9; :: thesis:
8419 = (67 * 125) + 44 ;
hence not 67 divides 8419 by NAT_4:9; :: thesis:
8419 = (71 * 118) + 41 ;
hence not 71 divides 8419 by NAT_4:9; :: thesis:
8419 = (73 * 115) + 24 ;
hence not 73 divides 8419 by NAT_4:9; :: thesis:
8419 = (79 * 106) + 45 ;
hence not 79 divides 8419 by NAT_4:9; :: thesis:
8419 = (83 * 101) + 36 ;
hence not 83 divides 8419 by NAT_4:9; :: thesis:
8419 = (89 * 94) + 53 ;
hence not 89 divides 8419 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8419 & n is prime holds
not n divides 8419 by XPRIMET1:48;
hence 8419 is prime by NAT_4:14; :: thesis: