7529 = (2 * 3764) + 1 ;
hence not 2 divides 7529 by NAT_4:9; :: thesis:
7529 = (3 * 2509) + 2 ;
hence not 3 divides 7529 by NAT_4:9; :: thesis:
7529 = (5 * 1505) + 4 ;
hence not 5 divides 7529 by NAT_4:9; :: thesis:
7529 = (7 * 1075) + 4 ;
hence not 7 divides 7529 by NAT_4:9; :: thesis:
7529 = (11 * 684) + 5 ;
hence not 11 divides 7529 by NAT_4:9; :: thesis:
7529 = (13 * 579) + 2 ;
hence not 13 divides 7529 by NAT_4:9; :: thesis:
7529 = (17 * 442) + 15 ;
hence not 17 divides 7529 by NAT_4:9; :: thesis:
7529 = (19 * 396) + 5 ;
hence not 19 divides 7529 by NAT_4:9; :: thesis:
7529 = (23 * 327) + 8 ;
hence not 23 divides 7529 by NAT_4:9; :: thesis:
7529 = (29 * 259) + 18 ;
hence not 29 divides 7529 by NAT_4:9; :: thesis:
7529 = (31 * 242) + 27 ;
hence not 31 divides 7529 by NAT_4:9; :: thesis:
7529 = (37 * 203) + 18 ;
hence not 37 divides 7529 by NAT_4:9; :: thesis:
7529 = (41 * 183) + 26 ;
hence not 41 divides 7529 by NAT_4:9; :: thesis:
7529 = (43 * 175) + 4 ;
hence not 43 divides 7529 by NAT_4:9; :: thesis:
7529 = (47 * 160) + 9 ;
hence not 47 divides 7529 by NAT_4:9; :: thesis:
7529 = (53 * 142) + 3 ;
hence not 53 divides 7529 by NAT_4:9; :: thesis:
7529 = (59 * 127) + 36 ;
hence not 59 divides 7529 by NAT_4:9; :: thesis:
7529 = (61 * 123) + 26 ;
hence not 61 divides 7529 by NAT_4:9; :: thesis:
7529 = (67 * 112) + 25 ;
hence not 67 divides 7529 by NAT_4:9; :: thesis:
7529 = (71 * 106) + 3 ;
hence not 71 divides 7529 by NAT_4:9; :: thesis:
7529 = (73 * 103) + 10 ;
hence not 73 divides 7529 by NAT_4:9; :: thesis:
7529 = (79 * 95) + 24 ;
hence not 79 divides 7529 by NAT_4:9; :: thesis:
7529 = (83 * 90) + 59 ;
hence not 83 divides 7529 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7529 & n is prime holds
not n divides 7529 by XPRIMET1:46;
hence 7529 is prime by NAT_4:14; :: thesis: