5011 = (2 * 2505) + 1 ;
hence not 2 divides 5011 by NAT_4:9; :: thesis:
5011 = (3 * 1670) + 1 ;
hence not 3 divides 5011 by NAT_4:9; :: thesis:
5011 = (5 * 1002) + 1 ;
hence not 5 divides 5011 by NAT_4:9; :: thesis:
5011 = (7 * 715) + 6 ;
hence not 7 divides 5011 by NAT_4:9; :: thesis:
5011 = (11 * 455) + 6 ;
hence not 11 divides 5011 by NAT_4:9; :: thesis:
5011 = (13 * 385) + 6 ;
hence not 13 divides 5011 by NAT_4:9; :: thesis:
5011 = (17 * 294) + 13 ;
hence not 17 divides 5011 by NAT_4:9; :: thesis:
5011 = (19 * 263) + 14 ;
hence not 19 divides 5011 by NAT_4:9; :: thesis:
5011 = (23 * 217) + 20 ;
hence not 23 divides 5011 by NAT_4:9; :: thesis:
5011 = (29 * 172) + 23 ;
hence not 29 divides 5011 by NAT_4:9; :: thesis:
5011 = (31 * 161) + 20 ;
hence not 31 divides 5011 by NAT_4:9; :: thesis:
5011 = (37 * 135) + 16 ;
hence not 37 divides 5011 by NAT_4:9; :: thesis:
5011 = (41 * 122) + 9 ;
hence not 41 divides 5011 by NAT_4:9; :: thesis:
5011 = (43 * 116) + 23 ;
hence not 43 divides 5011 by NAT_4:9; :: thesis:
5011 = (47 * 106) + 29 ;
hence not 47 divides 5011 by NAT_4:9; :: thesis:
5011 = (53 * 94) + 29 ;
hence not 53 divides 5011 by NAT_4:9; :: thesis:
5011 = (59 * 84) + 55 ;
hence not 59 divides 5011 by NAT_4:9; :: thesis:
5011 = (61 * 82) + 9 ;
hence not 61 divides 5011 by NAT_4:9; :: thesis:
5011 = (67 * 74) + 53 ;
hence not 67 divides 5011 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5011 & n is prime holds
not n divides 5011 by XPRIMET1:38;
hence 5011 is prime by NAT_4:14; :: thesis: