
theorem
  for f being real-valued FinSequence holds
    Mean (f ^ <*Mean f*>) = Mean f
  proof
    let f be real-valued FinSequence;
    Mean (f ^ <*Mean f*>)
        = (Sum f + Mean f) / len (f ^ <*Mean f*>) by RVSUM_1:74
       .= (Sum f + Mean f) / (len f + len <*Mean f*>) by FINSEQ_1:22
       .= (Sum f + Mean f) / (len f + 1) by FINSEQ_1:39
       .= ((len f) * (Mean f) + Mean f) / (len f + 1) by Huk1
       .= (Mean f) * (len f + 1) / (len f + 1)
       .= Mean f by XCMPLX_1:89;
    hence thesis;
  end;
