reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem Th52:
  for n holds Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
  = Partial_Sums(Rseq * seq).(n+1) - (Rseq * Partial_Sums(seq)).(n+1)
proof
  defpred P[Nat] means
Partial_Sums((Rseq - Rseq^\1) * Partial_Sums
(seq)).$1 = Partial_Sums(Rseq * seq).($1+1) - (Rseq * Partial_Sums(seq)).($1+1)
  ;
A1: Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 = ((Rseq - Rseq^\1)
  * Partial_Sums(seq)).0 by Def1
    .= (Rseq - Rseq^\1).0 * Partial_Sums(seq).0 by Def7
    .= (Rseq + -Rseq^\1).0 * seq.0 by Def1
    .= (Rseq.0 + (-Rseq^\1).0) * seq.0 by SEQ_1:7
    .= (Rseq.0 + -(Rseq^\1).0) * seq.0 by SEQ_1:10
    .= (Rseq.0 - (Rseq^\1).0) * seq.0
    .= (Rseq.0 - Rseq.(0+1)) * seq.0 by NAT_1:def 3
    .= Rseq.0 * seq.0 - Rseq.1 * seq.0 by RLVECT_1:35;
A2: (Rseq * Partial_Sums(seq)).(0+1) = Rseq.(0+1) * Partial_Sums(seq).(0+1)
  by Def7
    .= Rseq.(0+1) * (Partial_Sums(seq).0 + seq.(0+1)) by Def1
    .= Rseq.1 * (seq.0 + seq.1) by Def1
    .= Rseq.1 * seq.0 + Rseq.1 * seq.1 by RLVECT_1:def 5;
A3: now
    let n;
    assume P[n];
    then Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (Rseq *
    Partial_Sums(seq)).(n+1) = Partial_Sums(Rseq * seq).(n+1) - ((Rseq *
Partial_Sums(seq)).(n+1) - (Rseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:29;
    then
A4: Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (Rseq *
    Partial_Sums(seq)).(n+1) = Partial_Sums(Rseq * seq).(n+1) - 09(X) by
RLVECT_1:15;
    Partial_Sums(Rseq * seq).((n+1)+1) = Partial_Sums(Rseq * seq).(n+1) +
    (Rseq * seq).((n+1)+1) by Def1
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (Rseq *
    Partial_Sums(seq)).(n+1) + (Rseq * seq).((n+1)+1) by A4
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + ((Rseq *
    Partial_Sums(seq)).(n+1) + (Rseq * seq).((n+1)+1)) by RLVECT_1:def 3
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (Rseq.(n+1)
    * Partial_Sums(seq).(n+1) + (Rseq * seq).((n+1)+1)) by Def7
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq.(n+1
) - Rseq.((n+1)+1)) + Rseq.((n+1)+1)) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1
    ) * seq.((n+1)+1)) by Def7
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq.(n+1
) - Rseq.((n+1)+1)) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * Partial_Sums(
    seq).(n+1)) + Rseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 6
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq.(n+1
) - (Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * Partial_Sums(
    seq).(n+1)) + Rseq.((n+1)+1) * seq.((n+1)+1)) by NAT_1:def 3
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq.(n+1
) + -(Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * Partial_Sums
    (seq).(n+1)) + Rseq.((n+1)+1) * seq.((n+1)+1))
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq.(n+1
) + (-Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * Partial_Sums
    (seq).(n+1)) + Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQ_1:10
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (((Rseq - (
Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * Partial_Sums(seq).
    (n+1)) + Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQ_1:7
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + ((Rseq - (
Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + (Rseq.((n+1)+1) * Partial_Sums(seq)
    .(n+1) + Rseq.((n+1)+1) * seq.((n+1)+1))) by RLVECT_1:def 3
      .= (Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + (Rseq - (
Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1)) + (Rseq.((n+1)+1) * Partial_Sums(seq
    ).(n+1) + Rseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 3
      .= (Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n + ((Rseq -
Rseq^\1) * Partial_Sums(seq)).(n+1)) + (Rseq.((n+1)+1) * Partial_Sums(seq).(n+1
    ) + Rseq.((n+1)+1) * seq.((n+1)+1)) by Def7
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + (Rseq.((
    n+1)+1) * Partial_Sums(seq).(n+1) + Rseq.((n+1)+1) * seq.((n+1)+1)) by Def1
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + (Rseq.((
    n+1)+1) * (Partial_Sums(seq).(n+1) + seq.((n+1)+1))) by RLVECT_1:def 5
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + (Rseq.((
    n+1)+1) * Partial_Sums(seq).((n+1)+1)) by Def1
      .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + (Rseq *
    Partial_Sums(seq)).((n+1)+1) by Def7;
    then
    Partial_Sums(Rseq * seq).((n+1)+1) - (Rseq * Partial_Sums(seq)).((n+1
    ) +1) = Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + ((Rseq *
    Partial_Sums(seq)).((n+1)+1) - (Rseq * Partial_Sums(seq)).((n+1)+1)) by
RLVECT_1:def 3;
    then
    Partial_Sums(Rseq * seq).((n+1)+1) - (Rseq * Partial_Sums(seq)).((n+1
    ) +1) = Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1) + 09(X)
by RLVECT_1:15;
    hence P[n+1];
  end;
  Partial_Sums(Rseq * seq).(0+1) = Partial_Sums(Rseq * seq).0 + (Rseq *
  seq).(0+1) by Def1
    .= (Rseq * seq).0 + (Rseq * seq).1 by Def1
    .= Rseq.0 * seq.0 + (Rseq * seq).1 by Def7
    .= Rseq.0 * seq.0 + Rseq.1 * seq.1 by Def7;
  then
  Partial_Sums(Rseq * seq).(0+1) = (Rseq.0 * seq.0 + 09(X)) + Rseq.1 * seq
  .1
    .= (Rseq.0 * seq.0 + (Rseq.1 * seq.0 - Rseq.1 * seq.0)) + Rseq.1 * seq.1
  by RLVECT_1:15
    .= ((Rseq.0 * seq.0 + -(Rseq.1 * seq.0)) + Rseq.1 * seq.0) + Rseq.1 *
  seq.1 by RLVECT_1:def 3
    .= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 + (Rseq *
  Partial_Sums(seq)).(0+1) by A1,A2,RLVECT_1:def 3;
  then Partial_Sums(Rseq * seq).(0+1) - (Rseq * Partial_Sums(seq)).(0+1) =
  Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 + ((Rseq * Partial_Sums(
  seq)).(0+1) - (Rseq * Partial_Sums(seq)).(0+1)) by RLVECT_1:def 3;
  then Partial_Sums(Rseq * seq).(0+1) - (Rseq * Partial_Sums(seq)).(0+1) =
  Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 + 09(X) by RLVECT_1:15;
  then
A5: P[0];
  thus for n holds P[n] from NAT_1:sch 2(A5,A3);
end;
