reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th34:
  for X being set, b1, b2, b3 being real-valued ManySortedSet of X
  holds (b1+b2)+b3 = b1+(b2+b3)
proof
  let X be set, b1, b2, b3 be real-valued ManySortedSet of X;
  now
    let x be object;
    assume x in X;
    thus ((b1+b2)+b3).x = (b1+b2).x+b3.x by Def5
      .= b1.x+b2.x+b3.x by Def5
      .= b1.x+(b2.x+b3.x)
      .= b1.x+(b2+b3).x by Def5
      .= (b1+(b2+b3)).x by Def5;
  end;
  hence thesis;
end;
