1373 = (2 * 686) + 1 ;
hence not 2 divides 1373 by NAT_4:9; :: thesis:
1373 = (3 * 457) + 2 ;
hence not 3 divides 1373 by NAT_4:9; :: thesis:
1373 = (5 * 274) + 3 ;
hence not 5 divides 1373 by NAT_4:9; :: thesis:
1373 = (7 * 196) + 1 ;
hence not 7 divides 1373 by NAT_4:9; :: thesis:
1373 = (11 * 124) + 9 ;
hence not 11 divides 1373 by NAT_4:9; :: thesis:
1373 = (13 * 105) + 8 ;
hence not 13 divides 1373 by NAT_4:9; :: thesis:
1373 = (17 * 80) + 13 ;
hence not 17 divides 1373 by NAT_4:9; :: thesis:
1373 = (19 * 72) + 5 ;
hence not 19 divides 1373 by NAT_4:9; :: thesis:
1373 = (23 * 59) + 16 ;
hence not 23 divides 1373 by NAT_4:9; :: thesis:
1373 = (29 * 47) + 10 ;
hence not 29 divides 1373 by NAT_4:9; :: thesis:
1373 = (31 * 44) + 9 ;
hence not 31 divides 1373 by NAT_4:9; :: thesis:
1373 = (37 * 37) + 4 ;
hence not 37 divides 1373 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1373 & n is prime holds
not n divides 1373 by XPRIMET1:24;
hence 1373 is prime by NAT_4:14; :: thesis: