7253 = (2 * 3626) + 1 ;
hence not 2 divides 7253 by NAT_4:9; :: thesis:
7253 = (3 * 2417) + 2 ;
hence not 3 divides 7253 by NAT_4:9; :: thesis:
7253 = (5 * 1450) + 3 ;
hence not 5 divides 7253 by NAT_4:9; :: thesis:
7253 = (7 * 1036) + 1 ;
hence not 7 divides 7253 by NAT_4:9; :: thesis:
7253 = (11 * 659) + 4 ;
hence not 11 divides 7253 by NAT_4:9; :: thesis:
7253 = (13 * 557) + 12 ;
hence not 13 divides 7253 by NAT_4:9; :: thesis:
7253 = (17 * 426) + 11 ;
hence not 17 divides 7253 by NAT_4:9; :: thesis:
7253 = (19 * 381) + 14 ;
hence not 19 divides 7253 by NAT_4:9; :: thesis:
7253 = (23 * 315) + 8 ;
hence not 23 divides 7253 by NAT_4:9; :: thesis:
7253 = (29 * 250) + 3 ;
hence not 29 divides 7253 by NAT_4:9; :: thesis:
7253 = (31 * 233) + 30 ;
hence not 31 divides 7253 by NAT_4:9; :: thesis:
7253 = (37 * 196) + 1 ;
hence not 37 divides 7253 by NAT_4:9; :: thesis:
7253 = (41 * 176) + 37 ;
hence not 41 divides 7253 by NAT_4:9; :: thesis:
7253 = (43 * 168) + 29 ;
hence not 43 divides 7253 by NAT_4:9; :: thesis:
7253 = (47 * 154) + 15 ;
hence not 47 divides 7253 by NAT_4:9; :: thesis:
7253 = (53 * 136) + 45 ;
hence not 53 divides 7253 by NAT_4:9; :: thesis:
7253 = (59 * 122) + 55 ;
hence not 59 divides 7253 by NAT_4:9; :: thesis:
7253 = (61 * 118) + 55 ;
hence not 61 divides 7253 by NAT_4:9; :: thesis:
7253 = (67 * 108) + 17 ;
hence not 67 divides 7253 by NAT_4:9; :: thesis:
7253 = (71 * 102) + 11 ;
hence not 71 divides 7253 by NAT_4:9; :: thesis:
7253 = (73 * 99) + 26 ;
hence not 73 divides 7253 by NAT_4:9; :: thesis:
7253 = (79 * 91) + 64 ;
hence not 79 divides 7253 by NAT_4:9; :: thesis:
7253 = (83 * 87) + 32 ;
hence not 83 divides 7253 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7253 & n is prime holds
not n divides 7253 by XPRIMET1:46;
hence 7253 is prime by NAT_4:14; :: thesis: