2357 = (2 * 1178) + 1 ;
hence not 2 divides 2357 by NAT_4:9; :: thesis:
2357 = (3 * 785) + 2 ;
hence not 3 divides 2357 by NAT_4:9; :: thesis:
2357 = (5 * 471) + 2 ;
hence not 5 divides 2357 by NAT_4:9; :: thesis:
2357 = (7 * 336) + 5 ;
hence not 7 divides 2357 by NAT_4:9; :: thesis:
2357 = (11 * 214) + 3 ;
hence not 11 divides 2357 by NAT_4:9; :: thesis:
2357 = (13 * 181) + 4 ;
hence not 13 divides 2357 by NAT_4:9; :: thesis:
2357 = (17 * 138) + 11 ;
hence not 17 divides 2357 by NAT_4:9; :: thesis:
2357 = (19 * 124) + 1 ;
hence not 19 divides 2357 by NAT_4:9; :: thesis:
2357 = (23 * 102) + 11 ;
hence not 23 divides 2357 by NAT_4:9; :: thesis:
2357 = (29 * 81) + 8 ;
hence not 29 divides 2357 by NAT_4:9; :: thesis:
2357 = (31 * 76) + 1 ;
hence not 31 divides 2357 by NAT_4:9; :: thesis:
2357 = (37 * 63) + 26 ;
hence not 37 divides 2357 by NAT_4:9; :: thesis:
2357 = (41 * 57) + 20 ;
hence not 41 divides 2357 by NAT_4:9; :: thesis:
2357 = (43 * 54) + 35 ;
hence not 43 divides 2357 by NAT_4:9; :: thesis:
2357 = (47 * 50) + 7 ;
hence not 47 divides 2357 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2357 & n is prime holds
not n divides 2357 by XPRIMET1:30;
hence 2357 is prime by NAT_4:14; :: thesis: