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 Th52:
  for n holds Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n
  = Partial_Sums(Cseq * seq).(n+1) - (Cseq * Partial_Sums(seq)).(n+1)
proof
  defpred P[Nat] means
   Partial_Sums((Cseq - Cseq^\1) * Partial_Sums
(seq)).$1 = Partial_Sums(Cseq * seq).($1+1) - (Cseq * Partial_Sums(seq)).($1+1)
  ;
A1: Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 = ((Cseq - Cseq^\1)
  * Partial_Sums(seq)).0 by BHSP_4:def 1
    .= (Cseq - Cseq^\1).0 * Partial_Sums(seq).0 by Def8
    .= (Cseq + -Cseq^\1).0 * seq.0 by BHSP_4:def 1
    .= (Cseq.0 + (-Cseq^\1).0) * seq.0 by VALUED_1:1
    .= (Cseq.0 + -(Cseq^\1).0) * seq.0 by VALUED_1:8
    .= (Cseq.0 - (Cseq^\1).0) * seq.0
    .= (Cseq.0 - Cseq.(0+1)) * seq.0 by NAT_1:def 3
    .= Cseq.0 * seq.0 - Cseq.1 * seq.0 by CLVECT_1:10;
A2: (Cseq * Partial_Sums(seq)).(0+1) = Cseq.(0+1) * Partial_Sums(seq).(0+1)
  by Def8
    .= Cseq.(0+1) * (Partial_Sums(seq).0 + seq.(0+1)) by BHSP_4:def 1
    .= Cseq.1 * (seq.0 + seq.1) by BHSP_4:def 1
    .= Cseq.1 * seq.0 + Cseq.1 * seq.1 by CLVECT_1:def 2;
A3: now
    let n;
    assume P[n];
    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
A4: 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;
    Partial_Sums(Cseq * seq).((n+1)+1) = Partial_Sums(Cseq * seq).(n+1) +
    (Cseq * seq).((n+1)+1) by BHSP_4:def 1
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
    Partial_Sums(seq)).(n+1) + (Cseq * seq).((n+1)+1) by A4,RLVECT_1:13
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq *
    Partial_Sums(seq)).(n+1) + (Cseq * seq).((n+1)+1)) by RLVECT_1:def 3
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq.(n+1)
    * Partial_Sums(seq).(n+1) + (Cseq * seq).((n+1)+1)) by Def8
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ( ((Cseq.(n+
1)-Cseq.((n+1)+1))+Cseq.((n+1)+1)) *Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) *
    seq.((n+1)+1) ) by Def8
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ( ((Cseq.(n+
1) - Cseq.((n+1)+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(
    seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by CLVECT_1:def 3
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) - (Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(
    seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by NAT_1:def 3
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) + -(Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums
    (seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1))
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) + (-Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums
    (seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by VALUED_1:8
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(seq).
    (n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by VALUED_1:1
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + (Cseq.((n+1)+1) * Partial_Sums(seq)
    .(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1))) by RLVECT_1:def 3
      .= (Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1)) + (Cseq.((n+1)+1) * Partial_Sums(seq
    ).(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 3
      .= (Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq -
Cseq^\1) * Partial_Sums(seq)).(n+1)) + (Cseq.((n+1)+1) * Partial_Sums(seq).(n+1
    ) + Cseq.((n+1)+1) * seq.((n+1)+1)) by Def8
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
n+1)+1) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1)) by
BHSP_4:def 1
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
    n+1)+1) * (Partial_Sums(seq).(n+1) + seq.((n+1)+1))) by CLVECT_1:def 2
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
    n+1)+1) * Partial_Sums(seq).((n+1)+1)) by BHSP_4:def 1
      .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq *
    Partial_Sums(seq)).((n+1)+1) by Def8;
    then Partial_Sums(Cseq * seq).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+
    1)+1) = Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + ((Cseq *
    Partial_Sums(seq)).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+1)+1)) by
RLVECT_1:def 3;
    then Partial_Sums(Cseq * seq).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+
    1)+1) = Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + 09(X)
by RLVECT_1:15;
    hence P[n+1] by RLVECT_1:4;
  end;
  Partial_Sums(Cseq * seq).(0+1) = Partial_Sums(Cseq * seq).0 + (Cseq *
  seq).(0+1) by BHSP_4:def 1
    .= (Cseq * seq).0 + (Cseq * seq).1 by BHSP_4:def 1
    .= Cseq.0 * seq.0 + (Cseq * seq).1 by Def8
    .= Cseq.0 * seq.0 + Cseq.1 * seq.1 by Def8;
  then
  Partial_Sums(Cseq * seq).(0+1) = (Cseq.0 * seq.0 + 09(X)) + Cseq.1 * seq
  .1 by RLVECT_1:4
    .= (Cseq.0 * seq.0 + (Cseq.1 * seq.0 - Cseq.1 * seq.0)) + Cseq.1 * seq.1
  by RLVECT_1:15
    .= ((Cseq.0 * seq.0 + -(Cseq.1 * seq.0)) + Cseq.1 * seq.0) + Cseq.1 *
  seq.1 by RLVECT_1:def 3
    .= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + (Cseq *
  Partial_Sums(seq)).(0+1) by A1,A2,RLVECT_1:def 3;
  then Partial_Sums(Cseq * seq).(0+1) - (Cseq * Partial_Sums(seq)).(0+1) =
  Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + ((Cseq * Partial_Sums(
  seq)).(0+1) - (Cseq * Partial_Sums(seq)).(0+1)) by RLVECT_1:def 3;
  then Partial_Sums(Cseq * seq).(0+1) - (Cseq * Partial_Sums(seq)).(0+1) =
  Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + 09(X) by RLVECT_1:15;
  then
A5: P[0] by RLVECT_1:4;
  thus for n holds P[n] from NAT_1:sch 2(A5,A3);
end;
