3691 = (2 * 1845) + 1 ;
hence not 2 divides 3691 by NAT_4:9; :: thesis:
3691 = (3 * 1230) + 1 ;
hence not 3 divides 3691 by NAT_4:9; :: thesis:
3691 = (5 * 738) + 1 ;
hence not 5 divides 3691 by NAT_4:9; :: thesis:
3691 = (7 * 527) + 2 ;
hence not 7 divides 3691 by NAT_4:9; :: thesis:
3691 = (11 * 335) + 6 ;
hence not 11 divides 3691 by NAT_4:9; :: thesis:
3691 = (13 * 283) + 12 ;
hence not 13 divides 3691 by NAT_4:9; :: thesis:
3691 = (17 * 217) + 2 ;
hence not 17 divides 3691 by NAT_4:9; :: thesis:
3691 = (19 * 194) + 5 ;
hence not 19 divides 3691 by NAT_4:9; :: thesis:
3691 = (23 * 160) + 11 ;
hence not 23 divides 3691 by NAT_4:9; :: thesis:
3691 = (29 * 127) + 8 ;
hence not 29 divides 3691 by NAT_4:9; :: thesis:
3691 = (31 * 119) + 2 ;
hence not 31 divides 3691 by NAT_4:9; :: thesis:
3691 = (37 * 99) + 28 ;
hence not 37 divides 3691 by NAT_4:9; :: thesis:
3691 = (41 * 90) + 1 ;
hence not 41 divides 3691 by NAT_4:9; :: thesis:
3691 = (43 * 85) + 36 ;
hence not 43 divides 3691 by NAT_4:9; :: thesis:
3691 = (47 * 78) + 25 ;
hence not 47 divides 3691 by NAT_4:9; :: thesis:
3691 = (53 * 69) + 34 ;
hence not 53 divides 3691 by NAT_4:9; :: thesis:
3691 = (59 * 62) + 33 ;
hence not 59 divides 3691 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3691 & n is prime holds
not n divides 3691 by XPRIMET1:34;
hence 3691 is prime by NAT_4:14; :: thesis: