reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th17:
  r*Re(z)=Re(r*z) & r*Im(z)=Im(r*z)
proof
  reconsider r9 = r as Element of COMPLEX by XCMPLX_0:def 2;
  r = r +0*<i>;
  then
A1: Re r = r & Im r = 0 by COMPLEX1:12;
  r*z = Re r9 * Re z - Im r9 * Im z + (Re r9 * Im z + Re z * Im r9)*<i> by
COMPLEX1:82
    .= r * Re z+(r * Im z)*<i> by A1;
  hence thesis by COMPLEX1:12;
end;
