1117 = (2 * 558) + 1 ;
hence not 2 divides 1117 by NAT_4:9; :: thesis:
1117 = (3 * 372) + 1 ;
hence not 3 divides 1117 by NAT_4:9; :: thesis:
1117 = (5 * 223) + 2 ;
hence not 5 divides 1117 by NAT_4:9; :: thesis:
1117 = (7 * 159) + 4 ;
hence not 7 divides 1117 by NAT_4:9; :: thesis:
1117 = (11 * 101) + 6 ;
hence not 11 divides 1117 by NAT_4:9; :: thesis:
1117 = (13 * 85) + 12 ;
hence not 13 divides 1117 by NAT_4:9; :: thesis:
1117 = (17 * 65) + 12 ;
hence not 17 divides 1117 by NAT_4:9; :: thesis:
1117 = (19 * 58) + 15 ;
hence not 19 divides 1117 by NAT_4:9; :: thesis:
1117 = (23 * 48) + 13 ;
hence not 23 divides 1117 by NAT_4:9; :: thesis:
1117 = (29 * 38) + 15 ;
hence not 29 divides 1117 by NAT_4:9; :: thesis:
1117 = (31 * 36) + 1 ;
hence not 31 divides 1117 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1117 & n is prime holds
not n divides 1117 by XPRIMET1:22;
hence 1117 is prime by NAT_4:14; :: thesis: