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 Th53:
  for n holds Partial_Sums(Rseq * seq).(n+1) = (Rseq *
Partial_Sums(seq)).(n+1) - Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n
proof
  let n;
  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 Th52;
  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 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;
  then Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums(((-1) (#) (Rseq^\1) - -Rseq) * Partial_Sums(seq)).n;
  then Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums(((-1) (#) (Rseq^\1 - Rseq)) * Partial_Sums(seq)).n by SEQ_1:24;
  then Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums((-1) * ((Rseq^\1 - Rseq) * Partial_Sums(seq))).n by Th46;
  then
  Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) + ((-1
  ) * Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq))).n by Th3;
  then
  Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) + (-1)
  * Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n by NORMSP_1:def 5;
  hence thesis by RLVECT_1:16;
end;
