
theorem Th2:
  for V be Abelian add-associative right_zeroed non empty
  addLoopStr for p be FinSequence of the carrier of V holds Sum p = Sum Rev p
proof
  let V be Abelian add-associative right_zeroed non empty addLoopStr;
  defpred P[FinSequence of the carrier of V] means Sum $1 = Sum Rev $1;
A1: for p be FinSequence of the carrier of V for x be Element of V st P[p]
  holds P[p^<*x*>]
  proof
    let p be FinSequence of the carrier of V;
    let x be Element of V;
    assume
A2: Sum p = Sum Rev p;
    thus Sum (p^<*x*>) = Sum p + Sum <*x*> by RLVECT_1:41
      .= Sum (<*x*>^Rev p) by A2,RLVECT_1:41
      .= Sum Rev (p^<*x*>) by FINSEQ_5:63;
  end;
A3: P[<*>(the carrier of V)];
  thus for p be FinSequence of the carrier of V holds P[p] from FINSEQ_2:sch 2
  (A3,A1);
end;
