4261 = (2 * 2130) + 1 ;
hence not 2 divides 4261 by NAT_4:9; :: thesis:
4261 = (3 * 1420) + 1 ;
hence not 3 divides 4261 by NAT_4:9; :: thesis:
4261 = (5 * 852) + 1 ;
hence not 5 divides 4261 by NAT_4:9; :: thesis:
4261 = (7 * 608) + 5 ;
hence not 7 divides 4261 by NAT_4:9; :: thesis:
4261 = (11 * 387) + 4 ;
hence not 11 divides 4261 by NAT_4:9; :: thesis:
4261 = (13 * 327) + 10 ;
hence not 13 divides 4261 by NAT_4:9; :: thesis:
4261 = (17 * 250) + 11 ;
hence not 17 divides 4261 by NAT_4:9; :: thesis:
4261 = (19 * 224) + 5 ;
hence not 19 divides 4261 by NAT_4:9; :: thesis:
4261 = (23 * 185) + 6 ;
hence not 23 divides 4261 by NAT_4:9; :: thesis:
4261 = (29 * 146) + 27 ;
hence not 29 divides 4261 by NAT_4:9; :: thesis:
4261 = (31 * 137) + 14 ;
hence not 31 divides 4261 by NAT_4:9; :: thesis:
4261 = (37 * 115) + 6 ;
hence not 37 divides 4261 by NAT_4:9; :: thesis:
4261 = (41 * 103) + 38 ;
hence not 41 divides 4261 by NAT_4:9; :: thesis:
4261 = (43 * 99) + 4 ;
hence not 43 divides 4261 by NAT_4:9; :: thesis:
4261 = (47 * 90) + 31 ;
hence not 47 divides 4261 by NAT_4:9; :: thesis:
4261 = (53 * 80) + 21 ;
hence not 53 divides 4261 by NAT_4:9; :: thesis:
4261 = (59 * 72) + 13 ;
hence not 59 divides 4261 by NAT_4:9; :: thesis:
4261 = (61 * 69) + 52 ;
hence not 61 divides 4261 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4261 & n is prime holds
not n divides 4261 by XPRIMET1:36;
hence 4261 is prime by NAT_4:14; :: thesis: