6997 = (2 * 3498) + 1 ;
hence not 2 divides 6997 by NAT_4:9; :: thesis:
6997 = (3 * 2332) + 1 ;
hence not 3 divides 6997 by NAT_4:9; :: thesis:
6997 = (5 * 1399) + 2 ;
hence not 5 divides 6997 by NAT_4:9; :: thesis:
6997 = (7 * 999) + 4 ;
hence not 7 divides 6997 by NAT_4:9; :: thesis:
6997 = (11 * 636) + 1 ;
hence not 11 divides 6997 by NAT_4:9; :: thesis:
6997 = (13 * 538) + 3 ;
hence not 13 divides 6997 by NAT_4:9; :: thesis:
6997 = (17 * 411) + 10 ;
hence not 17 divides 6997 by NAT_4:9; :: thesis:
6997 = (19 * 368) + 5 ;
hence not 19 divides 6997 by NAT_4:9; :: thesis:
6997 = (23 * 304) + 5 ;
hence not 23 divides 6997 by NAT_4:9; :: thesis:
6997 = (29 * 241) + 8 ;
hence not 29 divides 6997 by NAT_4:9; :: thesis:
6997 = (31 * 225) + 22 ;
hence not 31 divides 6997 by NAT_4:9; :: thesis:
6997 = (37 * 189) + 4 ;
hence not 37 divides 6997 by NAT_4:9; :: thesis:
6997 = (41 * 170) + 27 ;
hence not 41 divides 6997 by NAT_4:9; :: thesis:
6997 = (43 * 162) + 31 ;
hence not 43 divides 6997 by NAT_4:9; :: thesis:
6997 = (47 * 148) + 41 ;
hence not 47 divides 6997 by NAT_4:9; :: thesis:
6997 = (53 * 132) + 1 ;
hence not 53 divides 6997 by NAT_4:9; :: thesis:
6997 = (59 * 118) + 35 ;
hence not 59 divides 6997 by NAT_4:9; :: thesis:
6997 = (61 * 114) + 43 ;
hence not 61 divides 6997 by NAT_4:9; :: thesis:
6997 = (67 * 104) + 29 ;
hence not 67 divides 6997 by NAT_4:9; :: thesis:
6997 = (71 * 98) + 39 ;
hence not 71 divides 6997 by NAT_4:9; :: thesis:
6997 = (73 * 95) + 62 ;
hence not 73 divides 6997 by NAT_4:9; :: thesis:
6997 = (79 * 88) + 45 ;
hence not 79 divides 6997 by NAT_4:9; :: thesis:
6997 = (83 * 84) + 25 ;
hence not 83 divides 6997 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6997 & n is prime holds
not n divides 6997 by XPRIMET1:46;
hence 6997 is prime by NAT_4:14; :: thesis: