1669 = (2 * 834) + 1 ;
hence not 2 divides 1669 by NAT_4:9; :: thesis:
1669 = (3 * 556) + 1 ;
hence not 3 divides 1669 by NAT_4:9; :: thesis:
1669 = (5 * 333) + 4 ;
hence not 5 divides 1669 by NAT_4:9; :: thesis:
1669 = (7 * 238) + 3 ;
hence not 7 divides 1669 by NAT_4:9; :: thesis:
1669 = (11 * 151) + 8 ;
hence not 11 divides 1669 by NAT_4:9; :: thesis:
1669 = (13 * 128) + 5 ;
hence not 13 divides 1669 by NAT_4:9; :: thesis:
1669 = (17 * 98) + 3 ;
hence not 17 divides 1669 by NAT_4:9; :: thesis:
1669 = (19 * 87) + 16 ;
hence not 19 divides 1669 by NAT_4:9; :: thesis:
1669 = (23 * 72) + 13 ;
hence not 23 divides 1669 by NAT_4:9; :: thesis:
1669 = (29 * 57) + 16 ;
hence not 29 divides 1669 by NAT_4:9; :: thesis:
1669 = (31 * 53) + 26 ;
hence not 31 divides 1669 by NAT_4:9; :: thesis:
1669 = (37 * 45) + 4 ;
hence not 37 divides 1669 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1669 & n is prime holds
not n divides 1669 by XPRIMET1:24;
hence 1669 is prime by NAT_4:14; :: thesis: