4013 = (2 * 2006) + 1 ;
hence not 2 divides 4013 by NAT_4:9; :: thesis:
4013 = (3 * 1337) + 2 ;
hence not 3 divides 4013 by NAT_4:9; :: thesis:
4013 = (5 * 802) + 3 ;
hence not 5 divides 4013 by NAT_4:9; :: thesis:
4013 = (7 * 573) + 2 ;
hence not 7 divides 4013 by NAT_4:9; :: thesis:
4013 = (11 * 364) + 9 ;
hence not 11 divides 4013 by NAT_4:9; :: thesis:
4013 = (13 * 308) + 9 ;
hence not 13 divides 4013 by NAT_4:9; :: thesis:
4013 = (17 * 236) + 1 ;
hence not 17 divides 4013 by NAT_4:9; :: thesis:
4013 = (19 * 211) + 4 ;
hence not 19 divides 4013 by NAT_4:9; :: thesis:
4013 = (23 * 174) + 11 ;
hence not 23 divides 4013 by NAT_4:9; :: thesis:
4013 = (29 * 138) + 11 ;
hence not 29 divides 4013 by NAT_4:9; :: thesis:
4013 = (31 * 129) + 14 ;
hence not 31 divides 4013 by NAT_4:9; :: thesis:
4013 = (37 * 108) + 17 ;
hence not 37 divides 4013 by NAT_4:9; :: thesis:
4013 = (41 * 97) + 36 ;
hence not 41 divides 4013 by NAT_4:9; :: thesis:
4013 = (43 * 93) + 14 ;
hence not 43 divides 4013 by NAT_4:9; :: thesis:
4013 = (47 * 85) + 18 ;
hence not 47 divides 4013 by NAT_4:9; :: thesis:
4013 = (53 * 75) + 38 ;
hence not 53 divides 4013 by NAT_4:9; :: thesis:
4013 = (59 * 68) + 1 ;
hence not 59 divides 4013 by NAT_4:9; :: thesis:
4013 = (61 * 65) + 48 ;
hence not 61 divides 4013 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4013 & n is prime holds
not n divides 4013 by XPRIMET1:36;
hence 4013 is prime by NAT_4:14; :: thesis: