
theorem AMB:
   for a,b be Nat holds a mod b <= a
   proof
     let a,b be Nat;
     per cases;
     suppose b = 0;
       hence thesis;
     end;
     suppose b <> 0; then
       reconsider b as non zero Nat;
       (a div b)*b + (a mod b) >= 0 + (a mod b) by XREAL_1:6;
       hence thesis by NAT_D:2;
     end;
   end;
