4451 = (2 * 2225) + 1 ;
hence not 2 divides 4451 by NAT_4:9; :: thesis:
4451 = (3 * 1483) + 2 ;
hence not 3 divides 4451 by NAT_4:9; :: thesis:
4451 = (5 * 890) + 1 ;
hence not 5 divides 4451 by NAT_4:9; :: thesis:
4451 = (7 * 635) + 6 ;
hence not 7 divides 4451 by NAT_4:9; :: thesis:
4451 = (11 * 404) + 7 ;
hence not 11 divides 4451 by NAT_4:9; :: thesis:
4451 = (13 * 342) + 5 ;
hence not 13 divides 4451 by NAT_4:9; :: thesis:
4451 = (17 * 261) + 14 ;
hence not 17 divides 4451 by NAT_4:9; :: thesis:
4451 = (19 * 234) + 5 ;
hence not 19 divides 4451 by NAT_4:9; :: thesis:
4451 = (23 * 193) + 12 ;
hence not 23 divides 4451 by NAT_4:9; :: thesis:
4451 = (29 * 153) + 14 ;
hence not 29 divides 4451 by NAT_4:9; :: thesis:
4451 = (31 * 143) + 18 ;
hence not 31 divides 4451 by NAT_4:9; :: thesis:
4451 = (37 * 120) + 11 ;
hence not 37 divides 4451 by NAT_4:9; :: thesis:
4451 = (41 * 108) + 23 ;
hence not 41 divides 4451 by NAT_4:9; :: thesis:
4451 = (43 * 103) + 22 ;
hence not 43 divides 4451 by NAT_4:9; :: thesis:
4451 = (47 * 94) + 33 ;
hence not 47 divides 4451 by NAT_4:9; :: thesis:
4451 = (53 * 83) + 52 ;
hence not 53 divides 4451 by NAT_4:9; :: thesis:
4451 = (59 * 75) + 26 ;
hence not 59 divides 4451 by NAT_4:9; :: thesis:
4451 = (61 * 72) + 59 ;
hence not 61 divides 4451 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4451 & n is prime holds
not n divides 4451 by XPRIMET1:36;
hence 4451 is prime by NAT_4:14; :: thesis: