7549 = (2 * 3774) + 1 ;
hence not 2 divides 7549 by NAT_4:9; :: thesis:
7549 = (3 * 2516) + 1 ;
hence not 3 divides 7549 by NAT_4:9; :: thesis:
7549 = (5 * 1509) + 4 ;
hence not 5 divides 7549 by NAT_4:9; :: thesis:
7549 = (7 * 1078) + 3 ;
hence not 7 divides 7549 by NAT_4:9; :: thesis:
7549 = (11 * 686) + 3 ;
hence not 11 divides 7549 by NAT_4:9; :: thesis:
7549 = (13 * 580) + 9 ;
hence not 13 divides 7549 by NAT_4:9; :: thesis:
7549 = (17 * 444) + 1 ;
hence not 17 divides 7549 by NAT_4:9; :: thesis:
7549 = (19 * 397) + 6 ;
hence not 19 divides 7549 by NAT_4:9; :: thesis:
7549 = (23 * 328) + 5 ;
hence not 23 divides 7549 by NAT_4:9; :: thesis:
7549 = (29 * 260) + 9 ;
hence not 29 divides 7549 by NAT_4:9; :: thesis:
7549 = (31 * 243) + 16 ;
hence not 31 divides 7549 by NAT_4:9; :: thesis:
7549 = (37 * 204) + 1 ;
hence not 37 divides 7549 by NAT_4:9; :: thesis:
7549 = (41 * 184) + 5 ;
hence not 41 divides 7549 by NAT_4:9; :: thesis:
7549 = (43 * 175) + 24 ;
hence not 43 divides 7549 by NAT_4:9; :: thesis:
7549 = (47 * 160) + 29 ;
hence not 47 divides 7549 by NAT_4:9; :: thesis:
7549 = (53 * 142) + 23 ;
hence not 53 divides 7549 by NAT_4:9; :: thesis:
7549 = (59 * 127) + 56 ;
hence not 59 divides 7549 by NAT_4:9; :: thesis:
7549 = (61 * 123) + 46 ;
hence not 61 divides 7549 by NAT_4:9; :: thesis:
7549 = (67 * 112) + 45 ;
hence not 67 divides 7549 by NAT_4:9; :: thesis:
7549 = (71 * 106) + 23 ;
hence not 71 divides 7549 by NAT_4:9; :: thesis:
7549 = (73 * 103) + 30 ;
hence not 73 divides 7549 by NAT_4:9; :: thesis:
7549 = (79 * 95) + 44 ;
hence not 79 divides 7549 by NAT_4:9; :: thesis:
7549 = (83 * 90) + 79 ;
hence not 83 divides 7549 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7549 & n is prime holds
not n divides 7549 by XPRIMET1:46;
hence 7549 is prime by NAT_4:14; :: thesis: