863 = (2 * 431) + 1 ;
hence not 2 divides 863 by NAT_4:9; :: thesis:
863 = (3 * 287) + 2 ;
hence not 3 divides 863 by NAT_4:9; :: thesis:
863 = (5 * 172) + 3 ;
hence not 5 divides 863 by NAT_4:9; :: thesis:
863 = (7 * 123) + 2 ;
hence not 7 divides 863 by NAT_4:9; :: thesis:
863 = (11 * 78) + 5 ;
hence not 11 divides 863 by NAT_4:9; :: thesis:
863 = (13 * 66) + 5 ;
hence not 13 divides 863 by NAT_4:9; :: thesis:
863 = (17 * 50) + 13 ;
hence not 17 divides 863 by NAT_4:9; :: thesis:
863 = (19 * 45) + 8 ;
hence not 19 divides 863 by NAT_4:9; :: thesis:
863 = (23 * 37) + 12 ;
hence not 23 divides 863 by NAT_4:9; :: thesis:
863 = (29 * 29) + 22 ;
hence not 29 divides 863 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 863 & n is prime holds
not n divides 863 by XPRIMET1:20;
hence 863 is prime by NAT_4:14; :: thesis: