2039 = (2 * 1019) + 1 ;
hence not 2 divides 2039 by NAT_4:9; :: thesis:
2039 = (3 * 679) + 2 ;
hence not 3 divides 2039 by NAT_4:9; :: thesis:
2039 = (5 * 407) + 4 ;
hence not 5 divides 2039 by NAT_4:9; :: thesis:
2039 = (7 * 291) + 2 ;
hence not 7 divides 2039 by NAT_4:9; :: thesis:
2039 = (11 * 185) + 4 ;
hence not 11 divides 2039 by NAT_4:9; :: thesis:
2039 = (13 * 156) + 11 ;
hence not 13 divides 2039 by NAT_4:9; :: thesis:
2039 = (17 * 119) + 16 ;
hence not 17 divides 2039 by NAT_4:9; :: thesis:
2039 = (19 * 107) + 6 ;
hence not 19 divides 2039 by NAT_4:9; :: thesis:
2039 = (23 * 88) + 15 ;
hence not 23 divides 2039 by NAT_4:9; :: thesis:
2039 = (29 * 70) + 9 ;
hence not 29 divides 2039 by NAT_4:9; :: thesis:
2039 = (31 * 65) + 24 ;
hence not 31 divides 2039 by NAT_4:9; :: thesis:
2039 = (37 * 55) + 4 ;
hence not 37 divides 2039 by NAT_4:9; :: thesis:
2039 = (41 * 49) + 30 ;
hence not 41 divides 2039 by NAT_4:9; :: thesis:
2039 = (43 * 47) + 18 ;
hence not 43 divides 2039 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2039 & n is prime holds
not n divides 2039 by XPRIMET1:28;
hence 2039 is prime by NAT_4:14; :: thesis: