
theorem HetMono:
  for f be real-valued FinSequence,
      i,j be Nat st i in dom f & j in dom f & i <> j &
        f.i <> Mean f & f.j <> Mean f holds
    Het f > Het Replace (f,i,j,Mean f,f.i + f.j - Mean f)
  proof
    let f be real-valued FinSequence,
        i,j be Nat;
    assume
A0: i in dom f & j in dom f & i <> j;
    assume
FF: f.i <> Mean f & f.j <> Mean f;
    set g = Replace (f,i,j,Mean f,f.i + f.j - Mean f);
zz: f,g are_gamma-equivalent by ReplaceGamma,A0;
    set a = Mean f;
    set b = f.i + f.j - Mean f;
FX: HetSet g <> HetSet f
    proof
      assume
h2:   HetSet g = HetSet f;
      not i in { n1 where n1 is Nat : n1 in dom g & g.n1 <> Mean g }
      proof
        assume i in { n1 where n1 is Nat : n1 in dom g & g.n1 <> Mean g };
        then consider n2 being Nat such that
G1:     n2 = i & n2 in dom g & g.n2 <> Mean g;
        thus thesis by zz,ReplaceValue3,A0,G1;
      end;
      hence thesis by h2,FF,A0;
    end;
    HetSet g c= HetSet f
    proof
      let x be object;
      assume x in HetSet g; then
      consider n1 being Nat such that
A1:   n1 = x & n1 in dom g & g.n1 <> Mean g;
A2:   n1 in dom f by A1,DinoDom;
      f.n1 <> Mean f
      proof
        per cases;
        suppose n1 = i;
          hence thesis by A1,zz,ReplaceValue3,A0;
        end;
        suppose
          n1 = j;
          hence thesis by FF;
        end;
        suppose
B1:       n1 <> i & n1 <> j;
          f, g are_gamma-equivalent by ReplaceGamma,A0;
          hence thesis by B1,ReplaceValue,A0,A2,A1;
        end;
      end;
      hence thesis by A1,A2;
    end; then
    HetSet g c< HetSet f by FX;
    hence thesis by CARD_2:48;
  end;
