 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);

theorem Th12:
  (multcomplexcpfunc A).(a,(multcomplexcpfunc A).(b,f))
  = (multcomplexcpfunc A).(a*b,f)
proof
  reconsider a,b as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider c = a*b as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider g = (multcomplexcpfunc A).(b,f) as Element of PFuncs(A,COMPLEX);
  reconsider h = (multcomplexcpfunc A).(a,g) as Element of PFuncs(A,COMPLEX);
  reconsider k = (multcomplexcpfunc A).(c,f) as Element of PFuncs(A,COMPLEX);
A1: dom h = dom g by Th7;
A2: dom g = dom f by Th7;
A3: now
    let x be Element of A;
    assume
A4: x in dom h;
    hence h.x =a*(g.x) by A1,Th7
      .=a*(b*(f.x)) by A2,A1,A4,Th7
      .=(a*b)*(f.x)
      .= k.x by A2,A1,A4,Th7;
  end;
  dom k = dom f by Th7;
  hence thesis by A2,A1,A3,PARTFUN1:5;
end;
