1493 = (2 * 746) + 1 ;
hence not 2 divides 1493 by NAT_4:9; :: thesis:
1493 = (3 * 497) + 2 ;
hence not 3 divides 1493 by NAT_4:9; :: thesis:
1493 = (5 * 298) + 3 ;
hence not 5 divides 1493 by NAT_4:9; :: thesis:
1493 = (7 * 213) + 2 ;
hence not 7 divides 1493 by NAT_4:9; :: thesis:
1493 = (11 * 135) + 8 ;
hence not 11 divides 1493 by NAT_4:9; :: thesis:
1493 = (13 * 114) + 11 ;
hence not 13 divides 1493 by NAT_4:9; :: thesis:
1493 = (17 * 87) + 14 ;
hence not 17 divides 1493 by NAT_4:9; :: thesis:
1493 = (19 * 78) + 11 ;
hence not 19 divides 1493 by NAT_4:9; :: thesis:
1493 = (23 * 64) + 21 ;
hence not 23 divides 1493 by NAT_4:9; :: thesis:
1493 = (29 * 51) + 14 ;
hence not 29 divides 1493 by NAT_4:9; :: thesis:
1493 = (31 * 48) + 5 ;
hence not 31 divides 1493 by NAT_4:9; :: thesis:
1493 = (37 * 40) + 13 ;
hence not 37 divides 1493 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1493 & n is prime holds
not n divides 1493 by XPRIMET1:24;
hence 1493 is prime by NAT_4:14; :: thesis: