5413 = (2 * 2706) + 1 ;
hence not 2 divides 5413 by NAT_4:9; :: thesis:
5413 = (3 * 1804) + 1 ;
hence not 3 divides 5413 by NAT_4:9; :: thesis:
5413 = (5 * 1082) + 3 ;
hence not 5 divides 5413 by NAT_4:9; :: thesis:
5413 = (7 * 773) + 2 ;
hence not 7 divides 5413 by NAT_4:9; :: thesis:
5413 = (11 * 492) + 1 ;
hence not 11 divides 5413 by NAT_4:9; :: thesis:
5413 = (13 * 416) + 5 ;
hence not 13 divides 5413 by NAT_4:9; :: thesis:
5413 = (17 * 318) + 7 ;
hence not 17 divides 5413 by NAT_4:9; :: thesis:
5413 = (19 * 284) + 17 ;
hence not 19 divides 5413 by NAT_4:9; :: thesis:
5413 = (23 * 235) + 8 ;
hence not 23 divides 5413 by NAT_4:9; :: thesis:
5413 = (29 * 186) + 19 ;
hence not 29 divides 5413 by NAT_4:9; :: thesis:
5413 = (31 * 174) + 19 ;
hence not 31 divides 5413 by NAT_4:9; :: thesis:
5413 = (37 * 146) + 11 ;
hence not 37 divides 5413 by NAT_4:9; :: thesis:
5413 = (41 * 132) + 1 ;
hence not 41 divides 5413 by NAT_4:9; :: thesis:
5413 = (43 * 125) + 38 ;
hence not 43 divides 5413 by NAT_4:9; :: thesis:
5413 = (47 * 115) + 8 ;
hence not 47 divides 5413 by NAT_4:9; :: thesis:
5413 = (53 * 102) + 7 ;
hence not 53 divides 5413 by NAT_4:9; :: thesis:
5413 = (59 * 91) + 44 ;
hence not 59 divides 5413 by NAT_4:9; :: thesis:
5413 = (61 * 88) + 45 ;
hence not 61 divides 5413 by NAT_4:9; :: thesis:
5413 = (67 * 80) + 53 ;
hence not 67 divides 5413 by NAT_4:9; :: thesis:
5413 = (71 * 76) + 17 ;
hence not 71 divides 5413 by NAT_4:9; :: thesis:
5413 = (73 * 74) + 11 ;
hence not 73 divides 5413 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5413 & n is prime holds
not n divides 5413 by XPRIMET1:42;
hence 5413 is prime by NAT_4:14; :: thesis: