3559 = (2 * 1779) + 1 ;
hence not 2 divides 3559 by NAT_4:9; :: thesis:
3559 = (3 * 1186) + 1 ;
hence not 3 divides 3559 by NAT_4:9; :: thesis:
3559 = (5 * 711) + 4 ;
hence not 5 divides 3559 by NAT_4:9; :: thesis:
3559 = (7 * 508) + 3 ;
hence not 7 divides 3559 by NAT_4:9; :: thesis:
3559 = (11 * 323) + 6 ;
hence not 11 divides 3559 by NAT_4:9; :: thesis:
3559 = (13 * 273) + 10 ;
hence not 13 divides 3559 by NAT_4:9; :: thesis:
3559 = (17 * 209) + 6 ;
hence not 17 divides 3559 by NAT_4:9; :: thesis:
3559 = (19 * 187) + 6 ;
hence not 19 divides 3559 by NAT_4:9; :: thesis:
3559 = (23 * 154) + 17 ;
hence not 23 divides 3559 by NAT_4:9; :: thesis:
3559 = (29 * 122) + 21 ;
hence not 29 divides 3559 by NAT_4:9; :: thesis:
3559 = (31 * 114) + 25 ;
hence not 31 divides 3559 by NAT_4:9; :: thesis:
3559 = (37 * 96) + 7 ;
hence not 37 divides 3559 by NAT_4:9; :: thesis:
3559 = (41 * 86) + 33 ;
hence not 41 divides 3559 by NAT_4:9; :: thesis:
3559 = (43 * 82) + 33 ;
hence not 43 divides 3559 by NAT_4:9; :: thesis:
3559 = (47 * 75) + 34 ;
hence not 47 divides 3559 by NAT_4:9; :: thesis:
3559 = (53 * 67) + 8 ;
hence not 53 divides 3559 by NAT_4:9; :: thesis:
3559 = (59 * 60) + 19 ;
hence not 59 divides 3559 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3559 & n is prime holds
not n divides 3559 by XPRIMET1:34;
hence 3559 is prime by NAT_4:14; :: thesis: