
theorem Th3:
  for p be REAL-valued FinSequence holds Sum p = Sum Rev p
proof
  let p be REAL-valued FinSequence;
A0: p is FinSequence of REAL by RVSUM_1:145;
  defpred P[FinSequence of REAL] means Sum $1 = Sum Rev $1;
A1: for p be FinSequence of REAL for x be Element of REAL st P[p] holds P[p^
  <*x*>]
  proof
    let p be FinSequence of REAL;
    let x be Element of REAL;
    assume
A2: Sum p = Sum Rev p;
    thus Sum (p^<*x*>) = Sum p + Sum <*x*> by RVSUM_1:75
      .= Sum (<*x*>^Rev p) by A2,RVSUM_1:75
      .= Sum Rev (p^<*x*>) by FINSEQ_5:63;
  end;
A3: P[<*>(REAL)];
  for p be FinSequence of REAL holds P[p] from FINSEQ_2:sch 2(A3,A1);
  hence thesis by A0;
end;
