3931 = (2 * 1965) + 1 ;
hence not 2 divides 3931 by NAT_4:9; :: thesis:
3931 = (3 * 1310) + 1 ;
hence not 3 divides 3931 by NAT_4:9; :: thesis:
3931 = (5 * 786) + 1 ;
hence not 5 divides 3931 by NAT_4:9; :: thesis:
3931 = (7 * 561) + 4 ;
hence not 7 divides 3931 by NAT_4:9; :: thesis:
3931 = (11 * 357) + 4 ;
hence not 11 divides 3931 by NAT_4:9; :: thesis:
3931 = (13 * 302) + 5 ;
hence not 13 divides 3931 by NAT_4:9; :: thesis:
3931 = (17 * 231) + 4 ;
hence not 17 divides 3931 by NAT_4:9; :: thesis:
3931 = (19 * 206) + 17 ;
hence not 19 divides 3931 by NAT_4:9; :: thesis:
3931 = (23 * 170) + 21 ;
hence not 23 divides 3931 by NAT_4:9; :: thesis:
3931 = (29 * 135) + 16 ;
hence not 29 divides 3931 by NAT_4:9; :: thesis:
3931 = (31 * 126) + 25 ;
hence not 31 divides 3931 by NAT_4:9; :: thesis:
3931 = (37 * 106) + 9 ;
hence not 37 divides 3931 by NAT_4:9; :: thesis:
3931 = (41 * 95) + 36 ;
hence not 41 divides 3931 by NAT_4:9; :: thesis:
3931 = (43 * 91) + 18 ;
hence not 43 divides 3931 by NAT_4:9; :: thesis:
3931 = (47 * 83) + 30 ;
hence not 47 divides 3931 by NAT_4:9; :: thesis:
3931 = (53 * 74) + 9 ;
hence not 53 divides 3931 by NAT_4:9; :: thesis:
3931 = (59 * 66) + 37 ;
hence not 59 divides 3931 by NAT_4:9; :: thesis:
3931 = (61 * 64) + 27 ;
hence not 61 divides 3931 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3931 & n is prime holds
not n divides 3931 by XPRIMET1:36;
hence 3931 is prime by NAT_4:14; :: thesis: