2543 = (2 * 1271) + 1 ;
hence not 2 divides 2543 by NAT_4:9; :: thesis:
2543 = (3 * 847) + 2 ;
hence not 3 divides 2543 by NAT_4:9; :: thesis:
2543 = (5 * 508) + 3 ;
hence not 5 divides 2543 by NAT_4:9; :: thesis:
2543 = (7 * 363) + 2 ;
hence not 7 divides 2543 by NAT_4:9; :: thesis:
2543 = (11 * 231) + 2 ;
hence not 11 divides 2543 by NAT_4:9; :: thesis:
2543 = (13 * 195) + 8 ;
hence not 13 divides 2543 by NAT_4:9; :: thesis:
2543 = (17 * 149) + 10 ;
hence not 17 divides 2543 by NAT_4:9; :: thesis:
2543 = (19 * 133) + 16 ;
hence not 19 divides 2543 by NAT_4:9; :: thesis:
2543 = (23 * 110) + 13 ;
hence not 23 divides 2543 by NAT_4:9; :: thesis:
2543 = (29 * 87) + 20 ;
hence not 29 divides 2543 by NAT_4:9; :: thesis:
2543 = (31 * 82) + 1 ;
hence not 31 divides 2543 by NAT_4:9; :: thesis:
2543 = (37 * 68) + 27 ;
hence not 37 divides 2543 by NAT_4:9; :: thesis:
2543 = (41 * 62) + 1 ;
hence not 41 divides 2543 by NAT_4:9; :: thesis:
2543 = (43 * 59) + 6 ;
hence not 43 divides 2543 by NAT_4:9; :: thesis:
2543 = (47 * 54) + 5 ;
hence not 47 divides 2543 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2543 & n is prime holds
not n divides 2543 by XPRIMET1:30;
hence 2543 is prime by NAT_4:14; :: thesis: