2203 = (2 * 1101) + 1 ;
hence not 2 divides 2203 by NAT_4:9; :: thesis:
2203 = (3 * 734) + 1 ;
hence not 3 divides 2203 by NAT_4:9; :: thesis:
2203 = (5 * 440) + 3 ;
hence not 5 divides 2203 by NAT_4:9; :: thesis:
2203 = (7 * 314) + 5 ;
hence not 7 divides 2203 by NAT_4:9; :: thesis:
2203 = (11 * 200) + 3 ;
hence not 11 divides 2203 by NAT_4:9; :: thesis:
2203 = (13 * 169) + 6 ;
hence not 13 divides 2203 by NAT_4:9; :: thesis:
2203 = (17 * 129) + 10 ;
hence not 17 divides 2203 by NAT_4:9; :: thesis:
2203 = (19 * 115) + 18 ;
hence not 19 divides 2203 by NAT_4:9; :: thesis:
2203 = (23 * 95) + 18 ;
hence not 23 divides 2203 by NAT_4:9; :: thesis:
2203 = (29 * 75) + 28 ;
hence not 29 divides 2203 by NAT_4:9; :: thesis:
2203 = (31 * 71) + 2 ;
hence not 31 divides 2203 by NAT_4:9; :: thesis:
2203 = (37 * 59) + 20 ;
hence not 37 divides 2203 by NAT_4:9; :: thesis:
2203 = (41 * 53) + 30 ;
hence not 41 divides 2203 by NAT_4:9; :: thesis:
2203 = (43 * 51) + 10 ;
hence not 43 divides 2203 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2203 & n is prime holds
not n divides 2203 by XPRIMET1:28;
hence 2203 is prime by NAT_4:14; :: thesis: