
theorem Th16:
  for i,j,n being Nat, a,b being bag of n
  holds (i,j)-cut (a+b) = (i,j)-cut(a) + (i,j)-cut(b)
proof
  let i,j,n be Nat, a,b be bag of n;
  set CUTAB = (i,j)-cut(a+b), CUTA = (i,j)-cut(a), CUTB=(i,j)-cut(b);
  now
    let x be object such that
A1: x in j-'i;
    j-'i = {k where k is Nat : k < j-'i } by AXIOMS:4;
    then ex k being Nat st ( k = x)&( k < j-'i) by A1;
    then reconsider x9 = x as Element of NAT by ORDINAL1:def 12;
    CUTAB.x = (a+b).(i+x9) by A1,Def1;
    then
A2: CUTAB.x = a.(i+x9) + b.(i+x9) by PRE_POLY:def 5;
A3: CUTA.x = a.(i+x9) by A1,Def1;
    CUTB.x = b.(i+x9) by A1,Def1;
    hence CUTAB.x = (CUTA + CUTB).x by A2,A3,PRE_POLY:def 5;
  end;
  hence thesis by PBOOLE:3;
end;
