1777 = (2 * 888) + 1 ;
hence not 2 divides 1777 by NAT_4:9; :: thesis:
1777 = (3 * 592) + 1 ;
hence not 3 divides 1777 by NAT_4:9; :: thesis:
1777 = (5 * 355) + 2 ;
hence not 5 divides 1777 by NAT_4:9; :: thesis:
1777 = (7 * 253) + 6 ;
hence not 7 divides 1777 by NAT_4:9; :: thesis:
1777 = (11 * 161) + 6 ;
hence not 11 divides 1777 by NAT_4:9; :: thesis:
1777 = (13 * 136) + 9 ;
hence not 13 divides 1777 by NAT_4:9; :: thesis:
1777 = (17 * 104) + 9 ;
hence not 17 divides 1777 by NAT_4:9; :: thesis:
1777 = (19 * 93) + 10 ;
hence not 19 divides 1777 by NAT_4:9; :: thesis:
1777 = (23 * 77) + 6 ;
hence not 23 divides 1777 by NAT_4:9; :: thesis:
1777 = (29 * 61) + 8 ;
hence not 29 divides 1777 by NAT_4:9; :: thesis:
1777 = (31 * 57) + 10 ;
hence not 31 divides 1777 by NAT_4:9; :: thesis:
1777 = (37 * 48) + 1 ;
hence not 37 divides 1777 by NAT_4:9; :: thesis:
1777 = (41 * 43) + 14 ;
hence not 41 divides 1777 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1777 & n is prime holds
not n divides 1777 by XPRIMET1:26;
hence 1777 is prime by NAT_4:14; :: thesis: