3187 = (2 * 1593) + 1 ;
hence not 2 divides 3187 by NAT_4:9; :: thesis:
3187 = (3 * 1062) + 1 ;
hence not 3 divides 3187 by NAT_4:9; :: thesis:
3187 = (5 * 637) + 2 ;
hence not 5 divides 3187 by NAT_4:9; :: thesis:
3187 = (7 * 455) + 2 ;
hence not 7 divides 3187 by NAT_4:9; :: thesis:
3187 = (11 * 289) + 8 ;
hence not 11 divides 3187 by NAT_4:9; :: thesis:
3187 = (13 * 245) + 2 ;
hence not 13 divides 3187 by NAT_4:9; :: thesis:
3187 = (17 * 187) + 8 ;
hence not 17 divides 3187 by NAT_4:9; :: thesis:
3187 = (19 * 167) + 14 ;
hence not 19 divides 3187 by NAT_4:9; :: thesis:
3187 = (23 * 138) + 13 ;
hence not 23 divides 3187 by NAT_4:9; :: thesis:
3187 = (29 * 109) + 26 ;
hence not 29 divides 3187 by NAT_4:9; :: thesis:
3187 = (31 * 102) + 25 ;
hence not 31 divides 3187 by NAT_4:9; :: thesis:
3187 = (37 * 86) + 5 ;
hence not 37 divides 3187 by NAT_4:9; :: thesis:
3187 = (41 * 77) + 30 ;
hence not 41 divides 3187 by NAT_4:9; :: thesis:
3187 = (43 * 74) + 5 ;
hence not 43 divides 3187 by NAT_4:9; :: thesis:
3187 = (47 * 67) + 38 ;
hence not 47 divides 3187 by NAT_4:9; :: thesis:
3187 = (53 * 60) + 7 ;
hence not 53 divides 3187 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3187 & n is prime holds
not n divides 3187 by XPRIMET1:32;
hence 3187 is prime by NAT_4:14; :: thesis: