1193 = (2 * 596) + 1 ;
hence not 2 divides 1193 by NAT_4:9; :: thesis:
1193 = (3 * 397) + 2 ;
hence not 3 divides 1193 by NAT_4:9; :: thesis:
1193 = (5 * 238) + 3 ;
hence not 5 divides 1193 by NAT_4:9; :: thesis:
1193 = (7 * 170) + 3 ;
hence not 7 divides 1193 by NAT_4:9; :: thesis:
1193 = (11 * 108) + 5 ;
hence not 11 divides 1193 by NAT_4:9; :: thesis:
1193 = (13 * 91) + 10 ;
hence not 13 divides 1193 by NAT_4:9; :: thesis:
1193 = (17 * 70) + 3 ;
hence not 17 divides 1193 by NAT_4:9; :: thesis:
1193 = (19 * 62) + 15 ;
hence not 19 divides 1193 by NAT_4:9; :: thesis:
1193 = (23 * 51) + 20 ;
hence not 23 divides 1193 by NAT_4:9; :: thesis:
1193 = (29 * 41) + 4 ;
hence not 29 divides 1193 by NAT_4:9; :: thesis:
1193 = (31 * 38) + 15 ;
hence not 31 divides 1193 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1193 & n is prime holds
not n divides 1193 by XPRIMET1:22;
hence 1193 is prime by NAT_4:14; :: thesis: