6949 = (2 * 3474) + 1 ;
hence not 2 divides 6949 by NAT_4:9; :: thesis:
6949 = (3 * 2316) + 1 ;
hence not 3 divides 6949 by NAT_4:9; :: thesis:
6949 = (5 * 1389) + 4 ;
hence not 5 divides 6949 by NAT_4:9; :: thesis:
6949 = (7 * 992) + 5 ;
hence not 7 divides 6949 by NAT_4:9; :: thesis:
6949 = (11 * 631) + 8 ;
hence not 11 divides 6949 by NAT_4:9; :: thesis:
6949 = (13 * 534) + 7 ;
hence not 13 divides 6949 by NAT_4:9; :: thesis:
6949 = (17 * 408) + 13 ;
hence not 17 divides 6949 by NAT_4:9; :: thesis:
6949 = (19 * 365) + 14 ;
hence not 19 divides 6949 by NAT_4:9; :: thesis:
6949 = (23 * 302) + 3 ;
hence not 23 divides 6949 by NAT_4:9; :: thesis:
6949 = (29 * 239) + 18 ;
hence not 29 divides 6949 by NAT_4:9; :: thesis:
6949 = (31 * 224) + 5 ;
hence not 31 divides 6949 by NAT_4:9; :: thesis:
6949 = (37 * 187) + 30 ;
hence not 37 divides 6949 by NAT_4:9; :: thesis:
6949 = (41 * 169) + 20 ;
hence not 41 divides 6949 by NAT_4:9; :: thesis:
6949 = (43 * 161) + 26 ;
hence not 43 divides 6949 by NAT_4:9; :: thesis:
6949 = (47 * 147) + 40 ;
hence not 47 divides 6949 by NAT_4:9; :: thesis:
6949 = (53 * 131) + 6 ;
hence not 53 divides 6949 by NAT_4:9; :: thesis:
6949 = (59 * 117) + 46 ;
hence not 59 divides 6949 by NAT_4:9; :: thesis:
6949 = (61 * 113) + 56 ;
hence not 61 divides 6949 by NAT_4:9; :: thesis:
6949 = (67 * 103) + 48 ;
hence not 67 divides 6949 by NAT_4:9; :: thesis:
6949 = (71 * 97) + 62 ;
hence not 71 divides 6949 by NAT_4:9; :: thesis:
6949 = (73 * 95) + 14 ;
hence not 73 divides 6949 by NAT_4:9; :: thesis:
6949 = (79 * 87) + 76 ;
hence not 79 divides 6949 by NAT_4:9; :: thesis:
6949 = (83 * 83) + 60 ;
hence not 83 divides 6949 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6949 & n is prime holds
not n divides 6949 by XPRIMET1:46;
hence 6949 is prime by NAT_4:14; :: thesis: