6961 = (2 * 3480) + 1 ;
hence not 2 divides 6961 by NAT_4:9; :: thesis:
6961 = (3 * 2320) + 1 ;
hence not 3 divides 6961 by NAT_4:9; :: thesis:
6961 = (5 * 1392) + 1 ;
hence not 5 divides 6961 by NAT_4:9; :: thesis:
6961 = (7 * 994) + 3 ;
hence not 7 divides 6961 by NAT_4:9; :: thesis:
6961 = (11 * 632) + 9 ;
hence not 11 divides 6961 by NAT_4:9; :: thesis:
6961 = (13 * 535) + 6 ;
hence not 13 divides 6961 by NAT_4:9; :: thesis:
6961 = (17 * 409) + 8 ;
hence not 17 divides 6961 by NAT_4:9; :: thesis:
6961 = (19 * 366) + 7 ;
hence not 19 divides 6961 by NAT_4:9; :: thesis:
6961 = (23 * 302) + 15 ;
hence not 23 divides 6961 by NAT_4:9; :: thesis:
6961 = (29 * 240) + 1 ;
hence not 29 divides 6961 by NAT_4:9; :: thesis:
6961 = (31 * 224) + 17 ;
hence not 31 divides 6961 by NAT_4:9; :: thesis:
6961 = (37 * 188) + 5 ;
hence not 37 divides 6961 by NAT_4:9; :: thesis:
6961 = (41 * 169) + 32 ;
hence not 41 divides 6961 by NAT_4:9; :: thesis:
6961 = (43 * 161) + 38 ;
hence not 43 divides 6961 by NAT_4:9; :: thesis:
6961 = (47 * 148) + 5 ;
hence not 47 divides 6961 by NAT_4:9; :: thesis:
6961 = (53 * 131) + 18 ;
hence not 53 divides 6961 by NAT_4:9; :: thesis:
6961 = (59 * 117) + 58 ;
hence not 59 divides 6961 by NAT_4:9; :: thesis:
6961 = (61 * 114) + 7 ;
hence not 61 divides 6961 by NAT_4:9; :: thesis:
6961 = (67 * 103) + 60 ;
hence not 67 divides 6961 by NAT_4:9; :: thesis:
6961 = (71 * 98) + 3 ;
hence not 71 divides 6961 by NAT_4:9; :: thesis:
6961 = (73 * 95) + 26 ;
hence not 73 divides 6961 by NAT_4:9; :: thesis:
6961 = (79 * 88) + 9 ;
hence not 79 divides 6961 by NAT_4:9; :: thesis:
6961 = (83 * 83) + 72 ;
hence not 83 divides 6961 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6961 & n is prime holds
not n divides 6961 by XPRIMET1:46;
hence 6961 is prime by NAT_4:14; :: thesis: