
theorem ExDiff:
  for f be positive heterogeneous non empty real-valued FinSequence
    ex i,j be Nat st
      i in dom f & j in dom f & i <> j & f.i < Mean f & Mean f < f.j
  proof
    let f be positive heterogeneous non empty real-valued FinSequence;
    take i = the Element of MeanLess f;
    take j = the Element of MeanMore f;
    i in MeanLess f; then
    consider ii being Nat such that
A1: ii = i & ii in dom f & f.ii < Mean f;
    j in MeanMore f; then
    consider jj being Nat such that
A2: jj = j & jj in dom f & f.jj > Mean f;
    thus thesis by A1,A2;
  end;
