1069 = (2 * 534) + 1 ;
hence not 2 divides 1069 by NAT_4:9; :: thesis:
1069 = (3 * 356) + 1 ;
hence not 3 divides 1069 by NAT_4:9; :: thesis:
1069 = (5 * 213) + 4 ;
hence not 5 divides 1069 by NAT_4:9; :: thesis:
1069 = (7 * 152) + 5 ;
hence not 7 divides 1069 by NAT_4:9; :: thesis:
1069 = (11 * 97) + 2 ;
hence not 11 divides 1069 by NAT_4:9; :: thesis:
1069 = (13 * 82) + 3 ;
hence not 13 divides 1069 by NAT_4:9; :: thesis:
1069 = (17 * 62) + 15 ;
hence not 17 divides 1069 by NAT_4:9; :: thesis:
1069 = (19 * 56) + 5 ;
hence not 19 divides 1069 by NAT_4:9; :: thesis:
1069 = (23 * 46) + 11 ;
hence not 23 divides 1069 by NAT_4:9; :: thesis:
1069 = (29 * 36) + 25 ;
hence not 29 divides 1069 by NAT_4:9; :: thesis:
1069 = (31 * 34) + 15 ;
hence not 31 divides 1069 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1069 & n is prime holds
not n divides 1069 by XPRIMET1:22;
hence 1069 is prime by NAT_4:14; :: thesis: