
theorem Th15:
  for i,j,n being Nat holds (i,j)-cut EmptyBag n = EmptyBag (j-'i)
proof
  let i,j,n be Nat;
  set CUT1 = (i,j)-cut EmptyBag n;
A1: dom CUT1 = j-'i by PARTFUN1:def 2;
  now
    let k be object;
    per cases;
    suppose
A2:   k in dom CUT1;
      j-'i = {x where x is Nat : x < j-'i} by AXIOMS:4;
      then ex x being Nat st ( k = x)&( x < j-'i) by A1,A2;
      then reconsider k9=k as Element of NAT by ORDINAL1:def 12;
      CUT1.k = (EmptyBag n).(i+k9) by A2,Def1
        .= 0 by PBOOLE:5;
      hence CUT1.k <= (EmptyBag (j-'i)).k;
    end;
    suppose not k in dom CUT1;
      hence CUT1.k <= (EmptyBag (j-'i)).k by FUNCT_1:def 2;
    end;
  end;
  then CUT1 divides EmptyBag (j-'i) by PRE_POLY:def 11;
  hence thesis by PRE_POLY:58;
end;
