
theorem Th8:
  for X being non empty TopSpace
  for f being continuous Function of the carrier of X,COMPLEX
      holds ( |.f.| is Function of the carrier of X,REAL &
         |.f.| is continuous)
proof
  let X be non empty TopSpace,
  f be continuous Function of the carrier of X,COMPLEX;
  reconsider h = |.f.| as Function of the carrier of X,REAL;
  for p being Point of X,V being Subset of REAL
      st h.p in V & V is open holds
        ex W being Subset of X st p in W & W is open & h.:W c= V
   proof
     let p be Point of X,V be Subset of REAL;
     assume
A1:    h.p in V & V is open;
     reconsider r0=h.p as Real;
     consider r being Real such that
A2:    0<r & ].r0-r,r0+r.[ c= V by A1,RCOMP_1:19;
    set S=].r0-r,r0+r.[;
    reconsider z1=f.p as Complex;
    set S1={y where y is Complex:|.(y-z1).| < r };
    S1 c= COMPLEX
    proof
      let z be object;
      assume z in S1;
      then ex y being Complex st
      z = y & |.(y - z1).| < r;
      hence z in COMPLEX by XCMPLX_0:def 2;
    end; then
    reconsider T1=S1 as Subset of COMPLEX;
A3:   T1 is open by CFDIFF_1:13;
    |.(z1 - z1).|=0;
    then z1 in S1 by A2;
    then consider W1 being Subset of X such that
A4: p in W1 & W1 is open & f.:W1 c= S1 by A3,Th3;
    set W=W1;
A5: W is open by A4;
A6: p in W by A4;
    h.:W c= ].r0-r,r0+r.[
    proof
        let x be object;
        assume x in h.:W;
        then consider z being object such that
A7:         z in dom h & z in W & h.z=x by FUNCT_1:def 6;
A8:     z in W1 by A7;
        reconsider pz=z as Point of X by A7;
        pz in the carrier of X;
        then pz in dom f by FUNCT_2:def 1;
        then f.pz in f.:W1 by A8,FUNCT_1:def 6;
        then
A9:      f.pz in S1 by A4;
        reconsider a1=f.pz as Complex;
        ex aa1 be Complex st f.pz = aa1 & |.(aa1-z1).| < r by A9;
        then
A10:      |.(a1 - z1).| < r;
A11:    |.(h.z - r0).| = |. |.(f.pz).| - (|.f.|).p .| by VALUED_1:18
                      .= |. |.(f.pz).| - |.(f.p).| .| by VALUED_1:18;
        |. |.(f.pz).| - |.(f.p).| .|
                    <= |. (f.pz) - (f.p).| by COMPLEX1:64;
        then |. |.(f.pz).| - |.(f.p).| .| < r by A10,XXREAL_0:2;
        hence x in S by A7,A11,RCOMP_1:1;
    end;
    hence ex W being Subset of X st p in W & W is open & h.:W c= V
                                        by A5,A6,A2,XBOOLE_1:1;
  end;
  hence thesis by C0SP2:1;
end;
