
theorem HomGMean:
  for f be heterogeneous positive non empty real-valued FinSequence holds
    GMean Homogen f > GMean f
  proof
    let f be heterogeneous positive non empty real-valued FinSequence;
    consider i,j being Nat such that
A1: i = the Element of MeanLess f &
    j = the Element of MeanMore f &
    Homogen f = Replace (f, i, j, Mean f, f.i + f.j - Mean f) by HomDef;
    i in dom f & j in dom f & i <> j by A1,BlaBla2; then
    consider g being positive non empty real-valued FinSequence such that
J1: g = Replace (f,i,j,Mean f,f.i + f.j - Mean f) &
      GMean f < GMean g by A1,ReplaceGMean3;
    thus thesis by J1,A1;
  end;
