2137 = (2 * 1068) + 1 ;
hence not 2 divides 2137 by NAT_4:9; :: thesis:
2137 = (3 * 712) + 1 ;
hence not 3 divides 2137 by NAT_4:9; :: thesis:
2137 = (5 * 427) + 2 ;
hence not 5 divides 2137 by NAT_4:9; :: thesis:
2137 = (7 * 305) + 2 ;
hence not 7 divides 2137 by NAT_4:9; :: thesis:
2137 = (11 * 194) + 3 ;
hence not 11 divides 2137 by NAT_4:9; :: thesis:
2137 = (13 * 164) + 5 ;
hence not 13 divides 2137 by NAT_4:9; :: thesis:
2137 = (17 * 125) + 12 ;
hence not 17 divides 2137 by NAT_4:9; :: thesis:
2137 = (19 * 112) + 9 ;
hence not 19 divides 2137 by NAT_4:9; :: thesis:
2137 = (23 * 92) + 21 ;
hence not 23 divides 2137 by NAT_4:9; :: thesis:
2137 = (29 * 73) + 20 ;
hence not 29 divides 2137 by NAT_4:9; :: thesis:
2137 = (31 * 68) + 29 ;
hence not 31 divides 2137 by NAT_4:9; :: thesis:
2137 = (37 * 57) + 28 ;
hence not 37 divides 2137 by NAT_4:9; :: thesis:
2137 = (41 * 52) + 5 ;
hence not 41 divides 2137 by NAT_4:9; :: thesis:
2137 = (43 * 49) + 30 ;
hence not 43 divides 2137 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2137 & n is prime holds
not n divides 2137 by XPRIMET1:28;
hence 2137 is prime by NAT_4:14; :: thesis: