4397 = (2 * 2198) + 1 ;
hence not 2 divides 4397 by NAT_4:9; :: thesis:
4397 = (3 * 1465) + 2 ;
hence not 3 divides 4397 by NAT_4:9; :: thesis:
4397 = (5 * 879) + 2 ;
hence not 5 divides 4397 by NAT_4:9; :: thesis:
4397 = (7 * 628) + 1 ;
hence not 7 divides 4397 by NAT_4:9; :: thesis:
4397 = (11 * 399) + 8 ;
hence not 11 divides 4397 by NAT_4:9; :: thesis:
4397 = (13 * 338) + 3 ;
hence not 13 divides 4397 by NAT_4:9; :: thesis:
4397 = (17 * 258) + 11 ;
hence not 17 divides 4397 by NAT_4:9; :: thesis:
4397 = (19 * 231) + 8 ;
hence not 19 divides 4397 by NAT_4:9; :: thesis:
4397 = (23 * 191) + 4 ;
hence not 23 divides 4397 by NAT_4:9; :: thesis:
4397 = (29 * 151) + 18 ;
hence not 29 divides 4397 by NAT_4:9; :: thesis:
4397 = (31 * 141) + 26 ;
hence not 31 divides 4397 by NAT_4:9; :: thesis:
4397 = (37 * 118) + 31 ;
hence not 37 divides 4397 by NAT_4:9; :: thesis:
4397 = (41 * 107) + 10 ;
hence not 41 divides 4397 by NAT_4:9; :: thesis:
4397 = (43 * 102) + 11 ;
hence not 43 divides 4397 by NAT_4:9; :: thesis:
4397 = (47 * 93) + 26 ;
hence not 47 divides 4397 by NAT_4:9; :: thesis:
4397 = (53 * 82) + 51 ;
hence not 53 divides 4397 by NAT_4:9; :: thesis:
4397 = (59 * 74) + 31 ;
hence not 59 divides 4397 by NAT_4:9; :: thesis:
4397 = (61 * 72) + 5 ;
hence not 61 divides 4397 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4397 & n is prime holds
not n divides 4397 by XPRIMET1:36;
hence 4397 is prime by NAT_4:14; :: thesis: