4441 = (2 * 2220) + 1 ;
hence not 2 divides 4441 by NAT_4:9; :: thesis:
4441 = (3 * 1480) + 1 ;
hence not 3 divides 4441 by NAT_4:9; :: thesis:
4441 = (5 * 888) + 1 ;
hence not 5 divides 4441 by NAT_4:9; :: thesis:
4441 = (7 * 634) + 3 ;
hence not 7 divides 4441 by NAT_4:9; :: thesis:
4441 = (11 * 403) + 8 ;
hence not 11 divides 4441 by NAT_4:9; :: thesis:
4441 = (13 * 341) + 8 ;
hence not 13 divides 4441 by NAT_4:9; :: thesis:
4441 = (17 * 261) + 4 ;
hence not 17 divides 4441 by NAT_4:9; :: thesis:
4441 = (19 * 233) + 14 ;
hence not 19 divides 4441 by NAT_4:9; :: thesis:
4441 = (23 * 193) + 2 ;
hence not 23 divides 4441 by NAT_4:9; :: thesis:
4441 = (29 * 153) + 4 ;
hence not 29 divides 4441 by NAT_4:9; :: thesis:
4441 = (31 * 143) + 8 ;
hence not 31 divides 4441 by NAT_4:9; :: thesis:
4441 = (37 * 120) + 1 ;
hence not 37 divides 4441 by NAT_4:9; :: thesis:
4441 = (41 * 108) + 13 ;
hence not 41 divides 4441 by NAT_4:9; :: thesis:
4441 = (43 * 103) + 12 ;
hence not 43 divides 4441 by NAT_4:9; :: thesis:
4441 = (47 * 94) + 23 ;
hence not 47 divides 4441 by NAT_4:9; :: thesis:
4441 = (53 * 83) + 42 ;
hence not 53 divides 4441 by NAT_4:9; :: thesis:
4441 = (59 * 75) + 16 ;
hence not 59 divides 4441 by NAT_4:9; :: thesis:
4441 = (61 * 72) + 49 ;
hence not 61 divides 4441 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4441 & n is prime holds
not n divides 4441 by XPRIMET1:36;
hence 4441 is prime by NAT_4:14; :: thesis: