3541 = (2 * 1770) + 1 ;
hence not 2 divides 3541 by NAT_4:9; :: thesis:
3541 = (3 * 1180) + 1 ;
hence not 3 divides 3541 by NAT_4:9; :: thesis:
3541 = (5 * 708) + 1 ;
hence not 5 divides 3541 by NAT_4:9; :: thesis:
3541 = (7 * 505) + 6 ;
hence not 7 divides 3541 by NAT_4:9; :: thesis:
3541 = (11 * 321) + 10 ;
hence not 11 divides 3541 by NAT_4:9; :: thesis:
3541 = (13 * 272) + 5 ;
hence not 13 divides 3541 by NAT_4:9; :: thesis:
3541 = (17 * 208) + 5 ;
hence not 17 divides 3541 by NAT_4:9; :: thesis:
3541 = (19 * 186) + 7 ;
hence not 19 divides 3541 by NAT_4:9; :: thesis:
3541 = (23 * 153) + 22 ;
hence not 23 divides 3541 by NAT_4:9; :: thesis:
3541 = (29 * 122) + 3 ;
hence not 29 divides 3541 by NAT_4:9; :: thesis:
3541 = (31 * 114) + 7 ;
hence not 31 divides 3541 by NAT_4:9; :: thesis:
3541 = (37 * 95) + 26 ;
hence not 37 divides 3541 by NAT_4:9; :: thesis:
3541 = (41 * 86) + 15 ;
hence not 41 divides 3541 by NAT_4:9; :: thesis:
3541 = (43 * 82) + 15 ;
hence not 43 divides 3541 by NAT_4:9; :: thesis:
3541 = (47 * 75) + 16 ;
hence not 47 divides 3541 by NAT_4:9; :: thesis:
3541 = (53 * 66) + 43 ;
hence not 53 divides 3541 by NAT_4:9; :: thesis:
3541 = (59 * 60) + 1 ;
hence not 59 divides 3541 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3541 & n is prime holds
not n divides 3541 by XPRIMET1:34;
hence 3541 is prime by NAT_4:14; :: thesis: