5419 = (2 * 2709) + 1 ;
hence not 2 divides 5419 by NAT_4:9; :: thesis:
5419 = (3 * 1806) + 1 ;
hence not 3 divides 5419 by NAT_4:9; :: thesis:
5419 = (5 * 1083) + 4 ;
hence not 5 divides 5419 by NAT_4:9; :: thesis:
5419 = (7 * 774) + 1 ;
hence not 7 divides 5419 by NAT_4:9; :: thesis:
5419 = (11 * 492) + 7 ;
hence not 11 divides 5419 by NAT_4:9; :: thesis:
5419 = (13 * 416) + 11 ;
hence not 13 divides 5419 by NAT_4:9; :: thesis:
5419 = (17 * 318) + 13 ;
hence not 17 divides 5419 by NAT_4:9; :: thesis:
5419 = (19 * 285) + 4 ;
hence not 19 divides 5419 by NAT_4:9; :: thesis:
5419 = (23 * 235) + 14 ;
hence not 23 divides 5419 by NAT_4:9; :: thesis:
5419 = (29 * 186) + 25 ;
hence not 29 divides 5419 by NAT_4:9; :: thesis:
5419 = (31 * 174) + 25 ;
hence not 31 divides 5419 by NAT_4:9; :: thesis:
5419 = (37 * 146) + 17 ;
hence not 37 divides 5419 by NAT_4:9; :: thesis:
5419 = (41 * 132) + 7 ;
hence not 41 divides 5419 by NAT_4:9; :: thesis:
5419 = (43 * 126) + 1 ;
hence not 43 divides 5419 by NAT_4:9; :: thesis:
5419 = (47 * 115) + 14 ;
hence not 47 divides 5419 by NAT_4:9; :: thesis:
5419 = (53 * 102) + 13 ;
hence not 53 divides 5419 by NAT_4:9; :: thesis:
5419 = (59 * 91) + 50 ;
hence not 59 divides 5419 by NAT_4:9; :: thesis:
5419 = (61 * 88) + 51 ;
hence not 61 divides 5419 by NAT_4:9; :: thesis:
5419 = (67 * 80) + 59 ;
hence not 67 divides 5419 by NAT_4:9; :: thesis:
5419 = (71 * 76) + 23 ;
hence not 71 divides 5419 by NAT_4:9; :: thesis:
5419 = (73 * 74) + 17 ;
hence not 73 divides 5419 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5419 & n is prime holds
not n divides 5419 by XPRIMET1:42;
hence 5419 is prime by NAT_4:14; :: thesis: