4091 = (2 * 2045) + 1 ;
hence not 2 divides 4091 by NAT_4:9; :: thesis:
4091 = (3 * 1363) + 2 ;
hence not 3 divides 4091 by NAT_4:9; :: thesis:
4091 = (5 * 818) + 1 ;
hence not 5 divides 4091 by NAT_4:9; :: thesis:
4091 = (7 * 584) + 3 ;
hence not 7 divides 4091 by NAT_4:9; :: thesis:
4091 = (11 * 371) + 10 ;
hence not 11 divides 4091 by NAT_4:9; :: thesis:
4091 = (13 * 314) + 9 ;
hence not 13 divides 4091 by NAT_4:9; :: thesis:
4091 = (17 * 240) + 11 ;
hence not 17 divides 4091 by NAT_4:9; :: thesis:
4091 = (19 * 215) + 6 ;
hence not 19 divides 4091 by NAT_4:9; :: thesis:
4091 = (23 * 177) + 20 ;
hence not 23 divides 4091 by NAT_4:9; :: thesis:
4091 = (29 * 141) + 2 ;
hence not 29 divides 4091 by NAT_4:9; :: thesis:
4091 = (31 * 131) + 30 ;
hence not 31 divides 4091 by NAT_4:9; :: thesis:
4091 = (37 * 110) + 21 ;
hence not 37 divides 4091 by NAT_4:9; :: thesis:
4091 = (41 * 99) + 32 ;
hence not 41 divides 4091 by NAT_4:9; :: thesis:
4091 = (43 * 95) + 6 ;
hence not 43 divides 4091 by NAT_4:9; :: thesis:
4091 = (47 * 87) + 2 ;
hence not 47 divides 4091 by NAT_4:9; :: thesis:
4091 = (53 * 77) + 10 ;
hence not 53 divides 4091 by NAT_4:9; :: thesis:
4091 = (59 * 69) + 20 ;
hence not 59 divides 4091 by NAT_4:9; :: thesis:
4091 = (61 * 67) + 4 ;
hence not 61 divides 4091 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4091 & n is prime holds
not n divides 4091 by XPRIMET1:36;
hence 4091 is prime by NAT_4:14; :: thesis: