7577 = (2 * 3788) + 1 ;
hence not 2 divides 7577 by NAT_4:9; :: thesis:
7577 = (3 * 2525) + 2 ;
hence not 3 divides 7577 by NAT_4:9; :: thesis:
7577 = (5 * 1515) + 2 ;
hence not 5 divides 7577 by NAT_4:9; :: thesis:
7577 = (7 * 1082) + 3 ;
hence not 7 divides 7577 by NAT_4:9; :: thesis:
7577 = (11 * 688) + 9 ;
hence not 11 divides 7577 by NAT_4:9; :: thesis:
7577 = (13 * 582) + 11 ;
hence not 13 divides 7577 by NAT_4:9; :: thesis:
7577 = (17 * 445) + 12 ;
hence not 17 divides 7577 by NAT_4:9; :: thesis:
7577 = (19 * 398) + 15 ;
hence not 19 divides 7577 by NAT_4:9; :: thesis:
7577 = (23 * 329) + 10 ;
hence not 23 divides 7577 by NAT_4:9; :: thesis:
7577 = (29 * 261) + 8 ;
hence not 29 divides 7577 by NAT_4:9; :: thesis:
7577 = (31 * 244) + 13 ;
hence not 31 divides 7577 by NAT_4:9; :: thesis:
7577 = (37 * 204) + 29 ;
hence not 37 divides 7577 by NAT_4:9; :: thesis:
7577 = (41 * 184) + 33 ;
hence not 41 divides 7577 by NAT_4:9; :: thesis:
7577 = (43 * 176) + 9 ;
hence not 43 divides 7577 by NAT_4:9; :: thesis:
7577 = (47 * 161) + 10 ;
hence not 47 divides 7577 by NAT_4:9; :: thesis:
7577 = (53 * 142) + 51 ;
hence not 53 divides 7577 by NAT_4:9; :: thesis:
7577 = (59 * 128) + 25 ;
hence not 59 divides 7577 by NAT_4:9; :: thesis:
7577 = (61 * 124) + 13 ;
hence not 61 divides 7577 by NAT_4:9; :: thesis:
7577 = (67 * 113) + 6 ;
hence not 67 divides 7577 by NAT_4:9; :: thesis:
7577 = (71 * 106) + 51 ;
hence not 71 divides 7577 by NAT_4:9; :: thesis:
7577 = (73 * 103) + 58 ;
hence not 73 divides 7577 by NAT_4:9; :: thesis:
7577 = (79 * 95) + 72 ;
hence not 79 divides 7577 by NAT_4:9; :: thesis:
7577 = (83 * 91) + 24 ;
hence not 83 divides 7577 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7577 & n is prime holds
not n divides 7577 by XPRIMET1:46;
hence 7577 is prime by NAT_4:14; :: thesis: