
theorem
  for f be positive heterogeneous non empty real-valued FinSequence
    for i,j be Nat st
    i in dom f & j in dom f & i <> j & i in MeanMore f & j in MeanLess f
      holds
    ex g being positive non empty real-valued FinSequence st
    g = Replace (f,i,j,Mean f,f.i + f.j - Mean f) &
      GMean f < GMean g
  proof
    let f be positive heterogeneous non empty real-valued FinSequence;
    let i, j be Nat such that
A1: i in dom f & j in dom f & i <> j & i in MeanMore f & j in MeanLess f;
    reconsider a = Mean f as positive Real;
b1: f.i > 0 & f.j > 0 by A1,PosDef;
h1: Mean f < f.i by A1,BlaBla2; then
h2: Mean f - Mean f < f.i - Mean f by XREAL_1:14;
    f.j > 0 by A1,PosDef; then
    f.i - Mean f + f.j > 0 by h2; then
    reconsider b = f.i + f.j - Mean f as positive Real;
    set g = Replace (f,i,j,a,b);
jj: dom f = dom g by DinoDom;
    reconsider g as positive non empty real-valued FinSequence
      by ReplacePositive,A1,h1;
    take g;
    f.i > Mean f + 0 & Mean f > f.j + 0 by A1,BlaBla2; then
    f.i - Mean f > 0 & Mean f - f.j > 0 by XREAL_1:20; then
    (f.i - Mean f) * (Mean f - f.j) > 0; then
    a * b - (f.i) * (f.j) + (f.i) * (f.j) > 0 + (f.i) * (f.j)
      by XREAL_1:6; then
    a * b / ((f.i) * (f.j)) > 1 by XREAL_1:187,b1; then
    (Product f) * (a * b / ((f.i) * (f.j))) > (Product f) * 1
      by XREAL_1:68; then
    (Product f) * (a * b) / ((f.i) * (f.j)) > Product f by XCMPLX_1:74; then
    (Product f) * a * b / ((f.i) * (f.j)) > Product f; then
    Product g > Product f by A1,ProdReplace;
    hence thesis by ProdGMean,jj,FINSEQ_3:29;
  end;
