
theorem Th5:
  for X being non empty TopSpace
  for a being Complex
  for f being continuous Function of the carrier of X,COMPLEX
   holds a(#)f is continuous Function of the carrier of X,COMPLEX
proof
  let X be non empty TopSpace;
  let a be Complex;
  let f be continuous Function of the carrier of X,COMPLEX;
  set h=a(#)f;
A1:for x be Point of X holds h.x=a*f.x by VALUED_1:6;
  now per cases;
  suppose
A2: a<>0;
    for p being Point of X,V being Subset of COMPLEX 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 COMPLEX;
      assume
A3:     h.p in V & V is open;
      reconsider z0=h.p as Complex;
      consider N being Neighbourhood of z0 such that
A4:     N c= V by A3,CFDIFF_1:9;
      consider r being Real such that
A5:   0<r & {y where y is Complex:|.(y-z0).| < r }c= N by CFDIFF_1:def 5;
      set S={y where y is Complex:|.(y-z0).| < r };
A6:  |.a.|>0 by A2;
A7:  r/|.a.|>0 by A2,A5;
      reconsider z1=f.p as Complex;
      set S1={y where y is Complex:|.(y-z1).| < r/|.a.| };
      S1 c= COMPLEX
      proof
        let z be object;
        assume z in S1;
        then ex y being Complex st
        z = y & |.(y - z1).| < r/|.a.|;
        hence z in COMPLEX by XCMPLX_0:def 2;
      end;
      then reconsider T1=S1 as Subset of COMPLEX;
A8:   T1 is open by CFDIFF_1:13;
      |.(z1 - z1).|=0;
      then z1 in S1 by A7;
      then consider W1 being Subset of X such that
A9:       p in W1 & W1 is open & f.:W1 c= S1 by A8,Th3;
      set W=W1;
A10:  W is open by A9;
A11:  p in W by A9;
      h.:W c= S
      proof
          let z3 be object;
          assume z3 in h.:W;
          then consider x3 being object such that
A12:          x3 in dom h & x3 in W & h.x3=z3 by FUNCT_1:def 6;
          reconsider px=x3 as Point of X by A12;
          px in the carrier of X;
          then px in dom f by FUNCT_2:def 1;
          then f.px in f.:W1 by A12,FUNCT_1:def 6;
          then
A13:        f.px in S1 by A9;
          reconsider a1=f.px as Complex;
          ex aa1 be Complex st f.px = aa1 & |.(aa1-z1).| < r/|.a.| by A13;
          then
A14:        |.(a1 - z1).| < r/|.a.|;
A15:      |.(h.x3 - z0).| = |.(a*f.px - z0).| by A1
                         .= |.(a*f.px  - a*f.p).| by A1
                         .= |.a*(f.px - f.p).|
                         .= |.a.| * |.(a1 - z1).| by COMPLEX1:65;
A16:      |.a.|*|.(a1 - z1).| <  |.a.|*(r/|.a.|) by A6,A14,XREAL_1:68;
          |.a.|*(r/|.a.|) = r*(|.a.|/|.a.|)
                         .= r*1 by A6,XCMPLX_1:60
                         .= r;
          then |.(h.px - z0).| < r by A15,A16;
          hence z3 in S by A12;
      end;
      then h.:W c= N by A5;
      hence ex W being Subset of X st p in W & W is open & h.:W c= V
                                        by A10,A11,A4,XBOOLE_1:1;
    end;
    hence a(#)f is continuous by Th3;
  end;
  suppose
A17: a=0;
    set g = X --> 0c;
    set CX=the carrier of X;
A18:dom g = CX by FUNCOP_1:13;
    dom h = CX by FUNCT_2:def 1;
    then
A19:  dom g = dom h by A18;
    for z be object st z in dom h holds g.z =h.z
    proof
      let z be object;
      assume
A20:    z in dom h;
      h.z = 0*(f.z) by A17,VALUED_1:6
         .= 0;
      hence thesis by A20,FUNCOP_1:7;
    end;
    hence a(#)f is continuous by A19,FUNCT_1:def 11;
  end;
  end;
  hence thesis;
end;
