4931 = (2 * 2465) + 1 ;
hence not 2 divides 4931 by NAT_4:9; :: thesis:
4931 = (3 * 1643) + 2 ;
hence not 3 divides 4931 by NAT_4:9; :: thesis:
4931 = (5 * 986) + 1 ;
hence not 5 divides 4931 by NAT_4:9; :: thesis:
4931 = (7 * 704) + 3 ;
hence not 7 divides 4931 by NAT_4:9; :: thesis:
4931 = (11 * 448) + 3 ;
hence not 11 divides 4931 by NAT_4:9; :: thesis:
4931 = (13 * 379) + 4 ;
hence not 13 divides 4931 by NAT_4:9; :: thesis:
4931 = (17 * 290) + 1 ;
hence not 17 divides 4931 by NAT_4:9; :: thesis:
4931 = (19 * 259) + 10 ;
hence not 19 divides 4931 by NAT_4:9; :: thesis:
4931 = (23 * 214) + 9 ;
hence not 23 divides 4931 by NAT_4:9; :: thesis:
4931 = (29 * 170) + 1 ;
hence not 29 divides 4931 by NAT_4:9; :: thesis:
4931 = (31 * 159) + 2 ;
hence not 31 divides 4931 by NAT_4:9; :: thesis:
4931 = (37 * 133) + 10 ;
hence not 37 divides 4931 by NAT_4:9; :: thesis:
4931 = (41 * 120) + 11 ;
hence not 41 divides 4931 by NAT_4:9; :: thesis:
4931 = (43 * 114) + 29 ;
hence not 43 divides 4931 by NAT_4:9; :: thesis:
4931 = (47 * 104) + 43 ;
hence not 47 divides 4931 by NAT_4:9; :: thesis:
4931 = (53 * 93) + 2 ;
hence not 53 divides 4931 by NAT_4:9; :: thesis:
4931 = (59 * 83) + 34 ;
hence not 59 divides 4931 by NAT_4:9; :: thesis:
4931 = (61 * 80) + 51 ;
hence not 61 divides 4931 by NAT_4:9; :: thesis:
4931 = (67 * 73) + 40 ;
hence not 67 divides 4931 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4931 & n is prime holds
not n divides 4931 by XPRIMET1:38;
hence 4931 is prime by NAT_4:14; :: thesis: