6991 = (2 * 3495) + 1 ;
hence not 2 divides 6991 by NAT_4:9; :: thesis:
6991 = (3 * 2330) + 1 ;
hence not 3 divides 6991 by NAT_4:9; :: thesis:
6991 = (5 * 1398) + 1 ;
hence not 5 divides 6991 by NAT_4:9; :: thesis:
6991 = (7 * 998) + 5 ;
hence not 7 divides 6991 by NAT_4:9; :: thesis:
6991 = (11 * 635) + 6 ;
hence not 11 divides 6991 by NAT_4:9; :: thesis:
6991 = (13 * 537) + 10 ;
hence not 13 divides 6991 by NAT_4:9; :: thesis:
6991 = (17 * 411) + 4 ;
hence not 17 divides 6991 by NAT_4:9; :: thesis:
6991 = (19 * 367) + 18 ;
hence not 19 divides 6991 by NAT_4:9; :: thesis:
6991 = (23 * 303) + 22 ;
hence not 23 divides 6991 by NAT_4:9; :: thesis:
6991 = (29 * 241) + 2 ;
hence not 29 divides 6991 by NAT_4:9; :: thesis:
6991 = (31 * 225) + 16 ;
hence not 31 divides 6991 by NAT_4:9; :: thesis:
6991 = (37 * 188) + 35 ;
hence not 37 divides 6991 by NAT_4:9; :: thesis:
6991 = (41 * 170) + 21 ;
hence not 41 divides 6991 by NAT_4:9; :: thesis:
6991 = (43 * 162) + 25 ;
hence not 43 divides 6991 by NAT_4:9; :: thesis:
6991 = (47 * 148) + 35 ;
hence not 47 divides 6991 by NAT_4:9; :: thesis:
6991 = (53 * 131) + 48 ;
hence not 53 divides 6991 by NAT_4:9; :: thesis:
6991 = (59 * 118) + 29 ;
hence not 59 divides 6991 by NAT_4:9; :: thesis:
6991 = (61 * 114) + 37 ;
hence not 61 divides 6991 by NAT_4:9; :: thesis:
6991 = (67 * 104) + 23 ;
hence not 67 divides 6991 by NAT_4:9; :: thesis:
6991 = (71 * 98) + 33 ;
hence not 71 divides 6991 by NAT_4:9; :: thesis:
6991 = (73 * 95) + 56 ;
hence not 73 divides 6991 by NAT_4:9; :: thesis:
6991 = (79 * 88) + 39 ;
hence not 79 divides 6991 by NAT_4:9; :: thesis:
6991 = (83 * 84) + 19 ;
hence not 83 divides 6991 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6991 & n is prime holds
not n divides 6991 by XPRIMET1:46;
hence 6991 is prime by NAT_4:14; :: thesis: