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 Th13:
  |.Re z.| <= |.z.| & |.Im z.| <= |.z.|
proof
  0 <= (Re z)^2 by XREAL_1:63;
  then
A1: 0+(Im z)^2 <= (Re z)^2 + (Im z)^2 by XREAL_1:6;
  0 <= (Im z)^2 by XREAL_1:63;
  then
A2: sqrt((Im z)^2) <= sqrt((Re z)^2 + (Im z)^2) by A1,SQUARE_1:26;
  0 <= (Im z)^2 by XREAL_1:63;
  then
A3: 0+(Re z)^2 <= (Im z)^2 + (Re z)^2 by XREAL_1:6;
  0 <= (Re z)^2 by XREAL_1:63;
  then |.z.| = sqrt ((Re z)^2 + (Im z)^2) & sqrt((Re z)^2) <= sqrt((Re z)^2 +
  (Im z )^2) by A3,COMPLEX1:def 12,SQUARE_1:26;
  hence thesis by A2,COMPLEX1:72;
end;
