2243 = (2 * 1121) + 1 ;
hence not 2 divides 2243 by NAT_4:9; :: thesis:
2243 = (3 * 747) + 2 ;
hence not 3 divides 2243 by NAT_4:9; :: thesis:
2243 = (5 * 448) + 3 ;
hence not 5 divides 2243 by NAT_4:9; :: thesis:
2243 = (7 * 320) + 3 ;
hence not 7 divides 2243 by NAT_4:9; :: thesis:
2243 = (11 * 203) + 10 ;
hence not 11 divides 2243 by NAT_4:9; :: thesis:
2243 = (13 * 172) + 7 ;
hence not 13 divides 2243 by NAT_4:9; :: thesis:
2243 = (17 * 131) + 16 ;
hence not 17 divides 2243 by NAT_4:9; :: thesis:
2243 = (19 * 118) + 1 ;
hence not 19 divides 2243 by NAT_4:9; :: thesis:
2243 = (23 * 97) + 12 ;
hence not 23 divides 2243 by NAT_4:9; :: thesis:
2243 = (29 * 77) + 10 ;
hence not 29 divides 2243 by NAT_4:9; :: thesis:
2243 = (31 * 72) + 11 ;
hence not 31 divides 2243 by NAT_4:9; :: thesis:
2243 = (37 * 60) + 23 ;
hence not 37 divides 2243 by NAT_4:9; :: thesis:
2243 = (41 * 54) + 29 ;
hence not 41 divides 2243 by NAT_4:9; :: thesis:
2243 = (43 * 52) + 7 ;
hence not 43 divides 2243 by NAT_4:9; :: thesis:
2243 = (47 * 47) + 34 ;
hence not 47 divides 2243 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2243 & n is prime holds
not n divides 2243 by XPRIMET1:30;
hence 2243 is prime by NAT_4:14; :: thesis: