7127 = (2 * 3563) + 1 ;
hence not 2 divides 7127 by NAT_4:9; :: thesis:
7127 = (3 * 2375) + 2 ;
hence not 3 divides 7127 by NAT_4:9; :: thesis:
7127 = (5 * 1425) + 2 ;
hence not 5 divides 7127 by NAT_4:9; :: thesis:
7127 = (7 * 1018) + 1 ;
hence not 7 divides 7127 by NAT_4:9; :: thesis:
7127 = (11 * 647) + 10 ;
hence not 11 divides 7127 by NAT_4:9; :: thesis:
7127 = (13 * 548) + 3 ;
hence not 13 divides 7127 by NAT_4:9; :: thesis:
7127 = (17 * 419) + 4 ;
hence not 17 divides 7127 by NAT_4:9; :: thesis:
7127 = (19 * 375) + 2 ;
hence not 19 divides 7127 by NAT_4:9; :: thesis:
7127 = (23 * 309) + 20 ;
hence not 23 divides 7127 by NAT_4:9; :: thesis:
7127 = (29 * 245) + 22 ;
hence not 29 divides 7127 by NAT_4:9; :: thesis:
7127 = (31 * 229) + 28 ;
hence not 31 divides 7127 by NAT_4:9; :: thesis:
7127 = (37 * 192) + 23 ;
hence not 37 divides 7127 by NAT_4:9; :: thesis:
7127 = (41 * 173) + 34 ;
hence not 41 divides 7127 by NAT_4:9; :: thesis:
7127 = (43 * 165) + 32 ;
hence not 43 divides 7127 by NAT_4:9; :: thesis:
7127 = (47 * 151) + 30 ;
hence not 47 divides 7127 by NAT_4:9; :: thesis:
7127 = (53 * 134) + 25 ;
hence not 53 divides 7127 by NAT_4:9; :: thesis:
7127 = (59 * 120) + 47 ;
hence not 59 divides 7127 by NAT_4:9; :: thesis:
7127 = (61 * 116) + 51 ;
hence not 61 divides 7127 by NAT_4:9; :: thesis:
7127 = (67 * 106) + 25 ;
hence not 67 divides 7127 by NAT_4:9; :: thesis:
7127 = (71 * 100) + 27 ;
hence not 71 divides 7127 by NAT_4:9; :: thesis:
7127 = (73 * 97) + 46 ;
hence not 73 divides 7127 by NAT_4:9; :: thesis:
7127 = (79 * 90) + 17 ;
hence not 79 divides 7127 by NAT_4:9; :: thesis:
7127 = (83 * 85) + 72 ;
hence not 83 divides 7127 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7127 & n is prime holds
not n divides 7127 by XPRIMET1:46;
hence 7127 is prime by NAT_4:14; :: thesis: