6977 = (2 * 3488) + 1 ;
hence not 2 divides 6977 by NAT_4:9; :: thesis:
6977 = (3 * 2325) + 2 ;
hence not 3 divides 6977 by NAT_4:9; :: thesis:
6977 = (5 * 1395) + 2 ;
hence not 5 divides 6977 by NAT_4:9; :: thesis:
6977 = (7 * 996) + 5 ;
hence not 7 divides 6977 by NAT_4:9; :: thesis:
6977 = (11 * 634) + 3 ;
hence not 11 divides 6977 by NAT_4:9; :: thesis:
6977 = (13 * 536) + 9 ;
hence not 13 divides 6977 by NAT_4:9; :: thesis:
6977 = (17 * 410) + 7 ;
hence not 17 divides 6977 by NAT_4:9; :: thesis:
6977 = (19 * 367) + 4 ;
hence not 19 divides 6977 by NAT_4:9; :: thesis:
6977 = (23 * 303) + 8 ;
hence not 23 divides 6977 by NAT_4:9; :: thesis:
6977 = (29 * 240) + 17 ;
hence not 29 divides 6977 by NAT_4:9; :: thesis:
6977 = (31 * 225) + 2 ;
hence not 31 divides 6977 by NAT_4:9; :: thesis:
6977 = (37 * 188) + 21 ;
hence not 37 divides 6977 by NAT_4:9; :: thesis:
6977 = (41 * 170) + 7 ;
hence not 41 divides 6977 by NAT_4:9; :: thesis:
6977 = (43 * 162) + 11 ;
hence not 43 divides 6977 by NAT_4:9; :: thesis:
6977 = (47 * 148) + 21 ;
hence not 47 divides 6977 by NAT_4:9; :: thesis:
6977 = (53 * 131) + 34 ;
hence not 53 divides 6977 by NAT_4:9; :: thesis:
6977 = (59 * 118) + 15 ;
hence not 59 divides 6977 by NAT_4:9; :: thesis:
6977 = (61 * 114) + 23 ;
hence not 61 divides 6977 by NAT_4:9; :: thesis:
6977 = (67 * 104) + 9 ;
hence not 67 divides 6977 by NAT_4:9; :: thesis:
6977 = (71 * 98) + 19 ;
hence not 71 divides 6977 by NAT_4:9; :: thesis:
6977 = (73 * 95) + 42 ;
hence not 73 divides 6977 by NAT_4:9; :: thesis:
6977 = (79 * 88) + 25 ;
hence not 79 divides 6977 by NAT_4:9; :: thesis:
6977 = (83 * 84) + 5 ;
hence not 83 divides 6977 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6977 & n is prime holds
not n divides 6977 by XPRIMET1:46;
hence 6977 is prime by NAT_4:14; :: thesis: