4007 = (2 * 2003) + 1 ;
hence not 2 divides 4007 by NAT_4:9; :: thesis:
4007 = (3 * 1335) + 2 ;
hence not 3 divides 4007 by NAT_4:9; :: thesis:
4007 = (5 * 801) + 2 ;
hence not 5 divides 4007 by NAT_4:9; :: thesis:
4007 = (7 * 572) + 3 ;
hence not 7 divides 4007 by NAT_4:9; :: thesis:
4007 = (11 * 364) + 3 ;
hence not 11 divides 4007 by NAT_4:9; :: thesis:
4007 = (13 * 308) + 3 ;
hence not 13 divides 4007 by NAT_4:9; :: thesis:
4007 = (17 * 235) + 12 ;
hence not 17 divides 4007 by NAT_4:9; :: thesis:
4007 = (19 * 210) + 17 ;
hence not 19 divides 4007 by NAT_4:9; :: thesis:
4007 = (23 * 174) + 5 ;
hence not 23 divides 4007 by NAT_4:9; :: thesis:
4007 = (29 * 138) + 5 ;
hence not 29 divides 4007 by NAT_4:9; :: thesis:
4007 = (31 * 129) + 8 ;
hence not 31 divides 4007 by NAT_4:9; :: thesis:
4007 = (37 * 108) + 11 ;
hence not 37 divides 4007 by NAT_4:9; :: thesis:
4007 = (41 * 97) + 30 ;
hence not 41 divides 4007 by NAT_4:9; :: thesis:
4007 = (43 * 93) + 8 ;
hence not 43 divides 4007 by NAT_4:9; :: thesis:
4007 = (47 * 85) + 12 ;
hence not 47 divides 4007 by NAT_4:9; :: thesis:
4007 = (53 * 75) + 32 ;
hence not 53 divides 4007 by NAT_4:9; :: thesis:
4007 = (59 * 67) + 54 ;
hence not 59 divides 4007 by NAT_4:9; :: thesis:
4007 = (61 * 65) + 42 ;
hence not 61 divides 4007 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4007 & n is prime holds
not n divides 4007 by XPRIMET1:36;
hence 4007 is prime by NAT_4:14; :: thesis: