7057 = (2 * 3528) + 1 ;
hence not 2 divides 7057 by NAT_4:9; :: thesis:
7057 = (3 * 2352) + 1 ;
hence not 3 divides 7057 by NAT_4:9; :: thesis:
7057 = (5 * 1411) + 2 ;
hence not 5 divides 7057 by NAT_4:9; :: thesis:
7057 = (7 * 1008) + 1 ;
hence not 7 divides 7057 by NAT_4:9; :: thesis:
7057 = (11 * 641) + 6 ;
hence not 11 divides 7057 by NAT_4:9; :: thesis:
7057 = (13 * 542) + 11 ;
hence not 13 divides 7057 by NAT_4:9; :: thesis:
7057 = (17 * 415) + 2 ;
hence not 17 divides 7057 by NAT_4:9; :: thesis:
7057 = (19 * 371) + 8 ;
hence not 19 divides 7057 by NAT_4:9; :: thesis:
7057 = (23 * 306) + 19 ;
hence not 23 divides 7057 by NAT_4:9; :: thesis:
7057 = (29 * 243) + 10 ;
hence not 29 divides 7057 by NAT_4:9; :: thesis:
7057 = (31 * 227) + 20 ;
hence not 31 divides 7057 by NAT_4:9; :: thesis:
7057 = (37 * 190) + 27 ;
hence not 37 divides 7057 by NAT_4:9; :: thesis:
7057 = (41 * 172) + 5 ;
hence not 41 divides 7057 by NAT_4:9; :: thesis:
7057 = (43 * 164) + 5 ;
hence not 43 divides 7057 by NAT_4:9; :: thesis:
7057 = (47 * 150) + 7 ;
hence not 47 divides 7057 by NAT_4:9; :: thesis:
7057 = (53 * 133) + 8 ;
hence not 53 divides 7057 by NAT_4:9; :: thesis:
7057 = (59 * 119) + 36 ;
hence not 59 divides 7057 by NAT_4:9; :: thesis:
7057 = (61 * 115) + 42 ;
hence not 61 divides 7057 by NAT_4:9; :: thesis:
7057 = (67 * 105) + 22 ;
hence not 67 divides 7057 by NAT_4:9; :: thesis:
7057 = (71 * 99) + 28 ;
hence not 71 divides 7057 by NAT_4:9; :: thesis:
7057 = (73 * 96) + 49 ;
hence not 73 divides 7057 by NAT_4:9; :: thesis:
7057 = (79 * 89) + 26 ;
hence not 79 divides 7057 by NAT_4:9; :: thesis:
7057 = (83 * 85) + 2 ;
hence not 83 divides 7057 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7057 & n is prime holds
not n divides 7057 by XPRIMET1:46;
hence 7057 is prime by NAT_4:14; :: thesis: