1553 = (2 * 776) + 1 ;
hence not 2 divides 1553 by NAT_4:9; :: thesis:
1553 = (3 * 517) + 2 ;
hence not 3 divides 1553 by NAT_4:9; :: thesis:
1553 = (5 * 310) + 3 ;
hence not 5 divides 1553 by NAT_4:9; :: thesis:
1553 = (7 * 221) + 6 ;
hence not 7 divides 1553 by NAT_4:9; :: thesis:
1553 = (11 * 141) + 2 ;
hence not 11 divides 1553 by NAT_4:9; :: thesis:
1553 = (13 * 119) + 6 ;
hence not 13 divides 1553 by NAT_4:9; :: thesis:
1553 = (17 * 91) + 6 ;
hence not 17 divides 1553 by NAT_4:9; :: thesis:
1553 = (19 * 81) + 14 ;
hence not 19 divides 1553 by NAT_4:9; :: thesis:
1553 = (23 * 67) + 12 ;
hence not 23 divides 1553 by NAT_4:9; :: thesis:
1553 = (29 * 53) + 16 ;
hence not 29 divides 1553 by NAT_4:9; :: thesis:
1553 = (31 * 50) + 3 ;
hence not 31 divides 1553 by NAT_4:9; :: thesis:
1553 = (37 * 41) + 36 ;
hence not 37 divides 1553 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1553 & n is prime holds
not n divides 1553 by XPRIMET1:24;
hence 1553 is prime by NAT_4:14; :: thesis: