
theorem WOWTheo:
  for f be heterogeneous positive non empty real-valued FinSequence holds
    ex g being non empty homogeneous positive real-valued FinSequence st
      GMean g > GMean f & Mean g = Mean f
  proof
    let f be heterogeneous positive non empty real-valued FinSequence;
    defpred P[Nat] means
      ex g being positive non empty real-valued FinSequence st
      Het g = $1 &
      Mean f = Mean g &
      GMean g > GMean f &
      Het g < Het f;
B1: ex k being Nat st P[k]
    proof
      reconsider g = Homogen f as positive non empty real-valued FinSequence;
      take k = Het g;
      take g;
      g, f are_gamma-equivalent by HomEqui;
      hence thesis by HomogenHet,HomGMean;
    end;
B2: for k being Nat st k <> 0 & P[k] ex n being Nat st n < k & P[n]
    proof
      let k be Nat;
      assume
Y1:   k <> 0 & P[k]; then
      consider g being positive non empty real-valued FinSequence such that
Y2:   Het g = k &
      Mean f = Mean g &
      GMean g > GMean f &
      Het g < Het f;
      reconsider g as heterogeneous
        positive non empty real-valued FinSequence by Y1,Y2,HetHetero;
      reconsider h = Homogen g as positive non empty real-valued FinSequence;
      take n = Het h;
      thus n < k by Y2,HomogenHet;
      thus P[n]
      proof
        ex g1 being positive non empty real-valued FinSequence st
        Het g1 = n &
        Mean f = Mean g1 &
        GMean g1 > GMean f &
        Het g1 < Het f
        proof
          take h;
          h, g are_gamma-equivalent by HomEqui;
          hence thesis by Y2,HomogenHet,XXREAL_0:2,HomGMean;
        end;
        hence thesis;
      end;
    end;
    P[0] from NAT_1:sch 7(B1,B2); then
    consider g being positive non empty real-valued FinSequence such that
WW:   Het g = 0 &
      Mean f = Mean g &
      GMean g > GMean f &
      Het g < Het f;
    g is homogeneous by WW;
    hence thesis by WW;
  end;
