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 Th12:
  |.z.| <= |.Re z.| + |.Im z.|
proof
  z = Re z + (Im z)*<i> by COMPLEX1:13;
  then
A1: |.z.| <= |.Re z +0*<i>.| +|.0+(Im z)*<i>.| by COMPLEX1:56;
  Re(0+(Im z)*<i>)=0 & Im(0+(Im z)*<i>)=Im z by COMPLEX1:12;
  hence thesis by A1,COMPLEX1:51;
end;
