1787 = (2 * 893) + 1 ;
hence not 2 divides 1787 by NAT_4:9; :: thesis:
1787 = (3 * 595) + 2 ;
hence not 3 divides 1787 by NAT_4:9; :: thesis:
1787 = (5 * 357) + 2 ;
hence not 5 divides 1787 by NAT_4:9; :: thesis:
1787 = (7 * 255) + 2 ;
hence not 7 divides 1787 by NAT_4:9; :: thesis:
1787 = (11 * 162) + 5 ;
hence not 11 divides 1787 by NAT_4:9; :: thesis:
1787 = (13 * 137) + 6 ;
hence not 13 divides 1787 by NAT_4:9; :: thesis:
1787 = (17 * 105) + 2 ;
hence not 17 divides 1787 by NAT_4:9; :: thesis:
1787 = (19 * 94) + 1 ;
hence not 19 divides 1787 by NAT_4:9; :: thesis:
1787 = (23 * 77) + 16 ;
hence not 23 divides 1787 by NAT_4:9; :: thesis:
1787 = (29 * 61) + 18 ;
hence not 29 divides 1787 by NAT_4:9; :: thesis:
1787 = (31 * 57) + 20 ;
hence not 31 divides 1787 by NAT_4:9; :: thesis:
1787 = (37 * 48) + 11 ;
hence not 37 divides 1787 by NAT_4:9; :: thesis:
1787 = (41 * 43) + 24 ;
hence not 41 divides 1787 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1787 & n is prime holds
not n divides 1787 by XPRIMET1:26;
hence 1787 is prime by NAT_4:14; :: thesis: