
theorem ::R2C:
  for rseq be Real_Sequence, cseq be Complex_Sequence st cseq = rseq
  holds Partial_Product rseq = Partial_Product cseq
  proof
    let rseq be Real_Sequence, cseq be Complex_Sequence such that
    A1: cseq = rseq;
    A3: dom (Partial_Product cseq) = NAT by COMSEQ_1:1
    .= dom (Partial_Product rseq) by SEQ_1:1;
    for k be Nat holds (Partial_Product cseq).k = (Partial_Product rseq).k
    proof
      let k be Nat;
      defpred P[Nat] means (Partial_Product cseq).$1 =
        (Partial_Product rseq).$1;
      B1: P[0]
      proof
        (Partial_Product cseq).0 = cseq.0 by PP
        .= (Partial_Product rseq).0 by A1,SERIES_3:def 1;
        hence thesis;
      end;
      B2: for j be Nat st P[j] holds P[j+1]
      proof
        let j be Nat such that
        C1: P[j];
        (Partial_Product cseq).(j+1) =
          (Partial_Product cseq).j * cseq.(j+1) by PP
        .= (Partial_Product rseq).(j+1) by A1,C1,SERIES_3:def 1;
        hence thesis;
      end;
      for n be Nat holds P[n] from NAT_1:sch 2(B1,B2);
      hence thesis;
    end;
    hence thesis by A3;
  end;
