3331 = (2 * 1665) + 1 ;
hence not 2 divides 3331 by NAT_4:9; :: thesis:
3331 = (3 * 1110) + 1 ;
hence not 3 divides 3331 by NAT_4:9; :: thesis:
3331 = (5 * 666) + 1 ;
hence not 5 divides 3331 by NAT_4:9; :: thesis:
3331 = (7 * 475) + 6 ;
hence not 7 divides 3331 by NAT_4:9; :: thesis:
3331 = (11 * 302) + 9 ;
hence not 11 divides 3331 by NAT_4:9; :: thesis:
3331 = (13 * 256) + 3 ;
hence not 13 divides 3331 by NAT_4:9; :: thesis:
3331 = (17 * 195) + 16 ;
hence not 17 divides 3331 by NAT_4:9; :: thesis:
3331 = (19 * 175) + 6 ;
hence not 19 divides 3331 by NAT_4:9; :: thesis:
3331 = (23 * 144) + 19 ;
hence not 23 divides 3331 by NAT_4:9; :: thesis:
3331 = (29 * 114) + 25 ;
hence not 29 divides 3331 by NAT_4:9; :: thesis:
3331 = (31 * 107) + 14 ;
hence not 31 divides 3331 by NAT_4:9; :: thesis:
3331 = (37 * 90) + 1 ;
hence not 37 divides 3331 by NAT_4:9; :: thesis:
3331 = (41 * 81) + 10 ;
hence not 41 divides 3331 by NAT_4:9; :: thesis:
3331 = (43 * 77) + 20 ;
hence not 43 divides 3331 by NAT_4:9; :: thesis:
3331 = (47 * 70) + 41 ;
hence not 47 divides 3331 by NAT_4:9; :: thesis:
3331 = (53 * 62) + 45 ;
hence not 53 divides 3331 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3331 & n is prime holds
not n divides 3331 by XPRIMET1:32;
hence 3331 is prime by NAT_4:14; :: thesis: