7523 = (2 * 3761) + 1 ;
hence not 2 divides 7523 by NAT_4:9; :: thesis:
7523 = (3 * 2507) + 2 ;
hence not 3 divides 7523 by NAT_4:9; :: thesis:
7523 = (5 * 1504) + 3 ;
hence not 5 divides 7523 by NAT_4:9; :: thesis:
7523 = (7 * 1074) + 5 ;
hence not 7 divides 7523 by NAT_4:9; :: thesis:
7523 = (11 * 683) + 10 ;
hence not 11 divides 7523 by NAT_4:9; :: thesis:
7523 = (13 * 578) + 9 ;
hence not 13 divides 7523 by NAT_4:9; :: thesis:
7523 = (17 * 442) + 9 ;
hence not 17 divides 7523 by NAT_4:9; :: thesis:
7523 = (19 * 395) + 18 ;
hence not 19 divides 7523 by NAT_4:9; :: thesis:
7523 = (23 * 327) + 2 ;
hence not 23 divides 7523 by NAT_4:9; :: thesis:
7523 = (29 * 259) + 12 ;
hence not 29 divides 7523 by NAT_4:9; :: thesis:
7523 = (31 * 242) + 21 ;
hence not 31 divides 7523 by NAT_4:9; :: thesis:
7523 = (37 * 203) + 12 ;
hence not 37 divides 7523 by NAT_4:9; :: thesis:
7523 = (41 * 183) + 20 ;
hence not 41 divides 7523 by NAT_4:9; :: thesis:
7523 = (43 * 174) + 41 ;
hence not 43 divides 7523 by NAT_4:9; :: thesis:
7523 = (47 * 160) + 3 ;
hence not 47 divides 7523 by NAT_4:9; :: thesis:
7523 = (53 * 141) + 50 ;
hence not 53 divides 7523 by NAT_4:9; :: thesis:
7523 = (59 * 127) + 30 ;
hence not 59 divides 7523 by NAT_4:9; :: thesis:
7523 = (61 * 123) + 20 ;
hence not 61 divides 7523 by NAT_4:9; :: thesis:
7523 = (67 * 112) + 19 ;
hence not 67 divides 7523 by NAT_4:9; :: thesis:
7523 = (71 * 105) + 68 ;
hence not 71 divides 7523 by NAT_4:9; :: thesis:
7523 = (73 * 103) + 4 ;
hence not 73 divides 7523 by NAT_4:9; :: thesis:
7523 = (79 * 95) + 18 ;
hence not 79 divides 7523 by NAT_4:9; :: thesis:
7523 = (83 * 90) + 53 ;
hence not 83 divides 7523 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7523 & n is prime holds
not n divides 7523 by XPRIMET1:46;
hence 7523 is prime by NAT_4:14; :: thesis: