1223 = (2 * 611) + 1 ;
hence not 2 divides 1223 by NAT_4:9; :: thesis:
1223 = (3 * 407) + 2 ;
hence not 3 divides 1223 by NAT_4:9; :: thesis:
1223 = (5 * 244) + 3 ;
hence not 5 divides 1223 by NAT_4:9; :: thesis:
1223 = (7 * 174) + 5 ;
hence not 7 divides 1223 by NAT_4:9; :: thesis:
1223 = (11 * 111) + 2 ;
hence not 11 divides 1223 by NAT_4:9; :: thesis:
1223 = (13 * 94) + 1 ;
hence not 13 divides 1223 by NAT_4:9; :: thesis:
1223 = (17 * 71) + 16 ;
hence not 17 divides 1223 by NAT_4:9; :: thesis:
1223 = (19 * 64) + 7 ;
hence not 19 divides 1223 by NAT_4:9; :: thesis:
1223 = (23 * 53) + 4 ;
hence not 23 divides 1223 by NAT_4:9; :: thesis:
1223 = (29 * 42) + 5 ;
hence not 29 divides 1223 by NAT_4:9; :: thesis:
1223 = (31 * 39) + 14 ;
hence not 31 divides 1223 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1223 & n is prime holds
not n divides 1223 by XPRIMET1:22;
hence 1223 is prime by NAT_4:14; :: thesis: