
theorem MeanSum:
  for f being real-valued FinSequence holds
    HetSet f = MeanLess f \/ MeanMore f
  proof
    let f be real-valued FinSequence;
    thus HetSet f c= MeanLess f \/ MeanMore f
    proof
      let x be object;
      assume x in HetSet f; then
      consider i being Nat such that
A1:   i = x & i in dom f & f.i <> Mean f;
      f.i < Mean f or f.i > Mean f by A1,XXREAL_0:1; then
      i in MeanLess f or i in MeanMore f by A1;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    let x be object;
    assume x in MeanLess f \/ MeanMore f; then
    per cases by XBOOLE_0:def 3;
    suppose x in MeanLess f; then
      ex i being Nat st i = x & i in dom f & f.i < Mean f;
      hence thesis;
    end;
    suppose x in MeanMore f; then
      ex i being Nat st i = x & i in dom f & f.i > Mean f;
      hence thesis;
    end;
  end;
