1789 = (2 * 894) + 1 ;
hence not 2 divides 1789 by NAT_4:9; :: thesis:
1789 = (3 * 596) + 1 ;
hence not 3 divides 1789 by NAT_4:9; :: thesis:
1789 = (5 * 357) + 4 ;
hence not 5 divides 1789 by NAT_4:9; :: thesis:
1789 = (7 * 255) + 4 ;
hence not 7 divides 1789 by NAT_4:9; :: thesis:
1789 = (11 * 162) + 7 ;
hence not 11 divides 1789 by NAT_4:9; :: thesis:
1789 = (13 * 137) + 8 ;
hence not 13 divides 1789 by NAT_4:9; :: thesis:
1789 = (17 * 105) + 4 ;
hence not 17 divides 1789 by NAT_4:9; :: thesis:
1789 = (19 * 94) + 3 ;
hence not 19 divides 1789 by NAT_4:9; :: thesis:
1789 = (23 * 77) + 18 ;
hence not 23 divides 1789 by NAT_4:9; :: thesis:
1789 = (29 * 61) + 20 ;
hence not 29 divides 1789 by NAT_4:9; :: thesis:
1789 = (31 * 57) + 22 ;
hence not 31 divides 1789 by NAT_4:9; :: thesis:
1789 = (37 * 48) + 13 ;
hence not 37 divides 1789 by NAT_4:9; :: thesis:
1789 = (41 * 43) + 26 ;
hence not 41 divides 1789 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1789 & n is prime holds
not n divides 1789 by XPRIMET1:26;
hence 1789 is prime by NAT_4:14; :: thesis: