5297 = (2 * 2648) + 1 ;
hence not 2 divides 5297 by NAT_4:9; :: thesis:
5297 = (3 * 1765) + 2 ;
hence not 3 divides 5297 by NAT_4:9; :: thesis:
5297 = (5 * 1059) + 2 ;
hence not 5 divides 5297 by NAT_4:9; :: thesis:
5297 = (7 * 756) + 5 ;
hence not 7 divides 5297 by NAT_4:9; :: thesis:
5297 = (11 * 481) + 6 ;
hence not 11 divides 5297 by NAT_4:9; :: thesis:
5297 = (13 * 407) + 6 ;
hence not 13 divides 5297 by NAT_4:9; :: thesis:
5297 = (17 * 311) + 10 ;
hence not 17 divides 5297 by NAT_4:9; :: thesis:
5297 = (19 * 278) + 15 ;
hence not 19 divides 5297 by NAT_4:9; :: thesis:
5297 = (23 * 230) + 7 ;
hence not 23 divides 5297 by NAT_4:9; :: thesis:
5297 = (29 * 182) + 19 ;
hence not 29 divides 5297 by NAT_4:9; :: thesis:
5297 = (31 * 170) + 27 ;
hence not 31 divides 5297 by NAT_4:9; :: thesis:
5297 = (37 * 143) + 6 ;
hence not 37 divides 5297 by NAT_4:9; :: thesis:
5297 = (41 * 129) + 8 ;
hence not 41 divides 5297 by NAT_4:9; :: thesis:
5297 = (43 * 123) + 8 ;
hence not 43 divides 5297 by NAT_4:9; :: thesis:
5297 = (47 * 112) + 33 ;
hence not 47 divides 5297 by NAT_4:9; :: thesis:
5297 = (53 * 99) + 50 ;
hence not 53 divides 5297 by NAT_4:9; :: thesis:
5297 = (59 * 89) + 46 ;
hence not 59 divides 5297 by NAT_4:9; :: thesis:
5297 = (61 * 86) + 51 ;
hence not 61 divides 5297 by NAT_4:9; :: thesis:
5297 = (67 * 79) + 4 ;
hence not 67 divides 5297 by NAT_4:9; :: thesis:
5297 = (71 * 74) + 43 ;
hence not 71 divides 5297 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5297 & n is prime holds
not n divides 5297 by XPRIMET1:40;
hence 5297 is prime by NAT_4:14; :: thesis: