877 = (2 * 438) + 1 ;
hence not 2 divides 877 by NAT_4:9; :: thesis:
877 = (3 * 292) + 1 ;
hence not 3 divides 877 by NAT_4:9; :: thesis:
877 = (5 * 175) + 2 ;
hence not 5 divides 877 by NAT_4:9; :: thesis:
877 = (7 * 125) + 2 ;
hence not 7 divides 877 by NAT_4:9; :: thesis:
877 = (11 * 79) + 8 ;
hence not 11 divides 877 by NAT_4:9; :: thesis:
877 = (13 * 67) + 6 ;
hence not 13 divides 877 by NAT_4:9; :: thesis:
877 = (17 * 51) + 10 ;
hence not 17 divides 877 by NAT_4:9; :: thesis:
877 = (19 * 46) + 3 ;
hence not 19 divides 877 by NAT_4:9; :: thesis:
877 = (23 * 38) + 3 ;
hence not 23 divides 877 by NAT_4:9; :: thesis:
877 = (29 * 30) + 7 ;
hence not 29 divides 877 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 877 & n is prime holds
not n divides 877 by XPRIMET1:20;
hence 877 is prime by NAT_4:14; :: thesis: