9431 = (2 * 4715) + 1 ;
hence not 2 divides 9431 by NAT_4:9; :: thesis:
9431 = (3 * 3143) + 2 ;
hence not 3 divides 9431 by NAT_4:9; :: thesis:
9431 = (5 * 1886) + 1 ;
hence not 5 divides 9431 by NAT_4:9; :: thesis:
9431 = (7 * 1347) + 2 ;
hence not 7 divides 9431 by NAT_4:9; :: thesis:
9431 = (11 * 857) + 4 ;
hence not 11 divides 9431 by NAT_4:9; :: thesis:
9431 = (13 * 725) + 6 ;
hence not 13 divides 9431 by NAT_4:9; :: thesis:
9431 = (17 * 554) + 13 ;
hence not 17 divides 9431 by NAT_4:9; :: thesis:
9431 = (19 * 496) + 7 ;
hence not 19 divides 9431 by NAT_4:9; :: thesis:
9431 = (23 * 410) + 1 ;
hence not 23 divides 9431 by NAT_4:9; :: thesis:
9431 = (29 * 325) + 6 ;
hence not 29 divides 9431 by NAT_4:9; :: thesis:
9431 = (31 * 304) + 7 ;
hence not 31 divides 9431 by NAT_4:9; :: thesis:
9431 = (37 * 254) + 33 ;
hence not 37 divides 9431 by NAT_4:9; :: thesis:
9431 = (41 * 230) + 1 ;
hence not 41 divides 9431 by NAT_4:9; :: thesis:
9431 = (43 * 219) + 14 ;
hence not 43 divides 9431 by NAT_4:9; :: thesis:
9431 = (47 * 200) + 31 ;
hence not 47 divides 9431 by NAT_4:9; :: thesis:
9431 = (53 * 177) + 50 ;
hence not 53 divides 9431 by NAT_4:9; :: thesis:
9431 = (59 * 159) + 50 ;
hence not 59 divides 9431 by NAT_4:9; :: thesis:
9431 = (61 * 154) + 37 ;
hence not 61 divides 9431 by NAT_4:9; :: thesis:
9431 = (67 * 140) + 51 ;
hence not 67 divides 9431 by NAT_4:9; :: thesis:
9431 = (71 * 132) + 59 ;
hence not 71 divides 9431 by NAT_4:9; :: thesis:
9431 = (73 * 129) + 14 ;
hence not 73 divides 9431 by NAT_4:9; :: thesis:
9431 = (79 * 119) + 30 ;
hence not 79 divides 9431 by NAT_4:9; :: thesis:
9431 = (83 * 113) + 52 ;
hence not 83 divides 9431 by NAT_4:9; :: thesis:
9431 = (89 * 105) + 86 ;
hence not 89 divides 9431 by NAT_4:9; :: thesis:
9431 = (97 * 97) + 22 ;
hence not 97 divides 9431 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9431 & n is prime holds
not n divides 9431 by XPRIMET1:50;
hence 9431 is prime by NAT_4:14; :: thesis: