1531 = (2 * 765) + 1 ;
hence not 2 divides 1531 by NAT_4:9; :: thesis:
1531 = (3 * 510) + 1 ;
hence not 3 divides 1531 by NAT_4:9; :: thesis:
1531 = (5 * 306) + 1 ;
hence not 5 divides 1531 by NAT_4:9; :: thesis:
1531 = (7 * 218) + 5 ;
hence not 7 divides 1531 by NAT_4:9; :: thesis:
1531 = (11 * 139) + 2 ;
hence not 11 divides 1531 by NAT_4:9; :: thesis:
1531 = (13 * 117) + 10 ;
hence not 13 divides 1531 by NAT_4:9; :: thesis:
1531 = (17 * 90) + 1 ;
hence not 17 divides 1531 by NAT_4:9; :: thesis:
1531 = (19 * 80) + 11 ;
hence not 19 divides 1531 by NAT_4:9; :: thesis:
1531 = (23 * 66) + 13 ;
hence not 23 divides 1531 by NAT_4:9; :: thesis:
1531 = (29 * 52) + 23 ;
hence not 29 divides 1531 by NAT_4:9; :: thesis:
1531 = (31 * 49) + 12 ;
hence not 31 divides 1531 by NAT_4:9; :: thesis:
1531 = (37 * 41) + 14 ;
hence not 37 divides 1531 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1531 & n is prime holds
not n divides 1531 by XPRIMET1:24;
hence 1531 is prime by NAT_4:14; :: thesis: