reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem Th53:
  for n holds Partial_Sums(Cseq * seq).(n+1) = (Cseq *
Partial_Sums(seq)).(n+1) - Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq)).n
proof
  let n;
  Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
  Partial_Sums(seq)).(n+1) = (Partial_Sums(Cseq * seq).(n+1) - (Cseq *
  Partial_Sums(seq)).(n+1)) + (Cseq * Partial_Sums(seq)).(n+1) by Th52;
  then Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
  Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - ((Cseq *
  Partial_Sums(seq)).(n+1) - (Cseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:29;
  then Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
  Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - 09(X) by
RLVECT_1:15;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums((Cseq + -(Cseq^\1))*Partial_Sums(seq)).n by RLVECT_1:13;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums(((-1r) (#) (Cseq^\1) - -Cseq) * Partial_Sums(seq)).n by
COMSEQ_1:11;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums(((-1r) (#) (Cseq^\1) - (-1r) (#) Cseq) * Partial_Sums(seq)).n
by COMSEQ_1:11;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums(((-1r) (#) (Cseq^\1 - Cseq)) * Partial_Sums(seq)).n by
COMSEQ_1:18;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
  Partial_Sums((-1r) * ((Cseq^\1 - Cseq) * Partial_Sums(seq))).n by Th46;
  then
  Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) + ((-
  1r) * Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq))).n by Th3;
  then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) + (-
  1r) * Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq)).n by CLVECT_1:def 14
;
  hence thesis by CLVECT_1:3;
end;
