7457 = (2 * 3728) + 1 ;
hence not 2 divides 7457 by NAT_4:9; :: thesis:
7457 = (3 * 2485) + 2 ;
hence not 3 divides 7457 by NAT_4:9; :: thesis:
7457 = (5 * 1491) + 2 ;
hence not 5 divides 7457 by NAT_4:9; :: thesis:
7457 = (7 * 1065) + 2 ;
hence not 7 divides 7457 by NAT_4:9; :: thesis:
7457 = (11 * 677) + 10 ;
hence not 11 divides 7457 by NAT_4:9; :: thesis:
7457 = (13 * 573) + 8 ;
hence not 13 divides 7457 by NAT_4:9; :: thesis:
7457 = (17 * 438) + 11 ;
hence not 17 divides 7457 by NAT_4:9; :: thesis:
7457 = (19 * 392) + 9 ;
hence not 19 divides 7457 by NAT_4:9; :: thesis:
7457 = (23 * 324) + 5 ;
hence not 23 divides 7457 by NAT_4:9; :: thesis:
7457 = (29 * 257) + 4 ;
hence not 29 divides 7457 by NAT_4:9; :: thesis:
7457 = (31 * 240) + 17 ;
hence not 31 divides 7457 by NAT_4:9; :: thesis:
7457 = (37 * 201) + 20 ;
hence not 37 divides 7457 by NAT_4:9; :: thesis:
7457 = (41 * 181) + 36 ;
hence not 41 divides 7457 by NAT_4:9; :: thesis:
7457 = (43 * 173) + 18 ;
hence not 43 divides 7457 by NAT_4:9; :: thesis:
7457 = (47 * 158) + 31 ;
hence not 47 divides 7457 by NAT_4:9; :: thesis:
7457 = (53 * 140) + 37 ;
hence not 53 divides 7457 by NAT_4:9; :: thesis:
7457 = (59 * 126) + 23 ;
hence not 59 divides 7457 by NAT_4:9; :: thesis:
7457 = (61 * 122) + 15 ;
hence not 61 divides 7457 by NAT_4:9; :: thesis:
7457 = (67 * 111) + 20 ;
hence not 67 divides 7457 by NAT_4:9; :: thesis:
7457 = (71 * 105) + 2 ;
hence not 71 divides 7457 by NAT_4:9; :: thesis:
7457 = (73 * 102) + 11 ;
hence not 73 divides 7457 by NAT_4:9; :: thesis:
7457 = (79 * 94) + 31 ;
hence not 79 divides 7457 by NAT_4:9; :: thesis:
7457 = (83 * 89) + 70 ;
hence not 83 divides 7457 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7457 & n is prime holds
not n divides 7457 by XPRIMET1:46;
hence 7457 is prime by NAT_4:14; :: thesis: