3511 = (2 * 1755) + 1 ;
hence not 2 divides 3511 by NAT_4:9; :: thesis:
3511 = (3 * 1170) + 1 ;
hence not 3 divides 3511 by NAT_4:9; :: thesis:
3511 = (5 * 702) + 1 ;
hence not 5 divides 3511 by NAT_4:9; :: thesis:
3511 = (7 * 501) + 4 ;
hence not 7 divides 3511 by NAT_4:9; :: thesis:
3511 = (11 * 319) + 2 ;
hence not 11 divides 3511 by NAT_4:9; :: thesis:
3511 = (13 * 270) + 1 ;
hence not 13 divides 3511 by NAT_4:9; :: thesis:
3511 = (17 * 206) + 9 ;
hence not 17 divides 3511 by NAT_4:9; :: thesis:
3511 = (19 * 184) + 15 ;
hence not 19 divides 3511 by NAT_4:9; :: thesis:
3511 = (23 * 152) + 15 ;
hence not 23 divides 3511 by NAT_4:9; :: thesis:
3511 = (29 * 121) + 2 ;
hence not 29 divides 3511 by NAT_4:9; :: thesis:
3511 = (31 * 113) + 8 ;
hence not 31 divides 3511 by NAT_4:9; :: thesis:
3511 = (37 * 94) + 33 ;
hence not 37 divides 3511 by NAT_4:9; :: thesis:
3511 = (41 * 85) + 26 ;
hence not 41 divides 3511 by NAT_4:9; :: thesis:
3511 = (43 * 81) + 28 ;
hence not 43 divides 3511 by NAT_4:9; :: thesis:
3511 = (47 * 74) + 33 ;
hence not 47 divides 3511 by NAT_4:9; :: thesis:
3511 = (53 * 66) + 13 ;
hence not 53 divides 3511 by NAT_4:9; :: thesis:
3511 = (59 * 59) + 30 ;
hence not 59 divides 3511 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3511 & n is prime holds
not n divides 3511 by XPRIMET1:34;
hence 3511 is prime by NAT_4:14; :: thesis: