 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 Th13:
  (addcpfunc A).((multcomplexcpfunc A).(a,f),(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).(a,f) as Element of PFuncs(A,COMPLEX);
  reconsider h = (multcomplexcpfunc A).(b,f) as Element of PFuncs(A,COMPLEX);
  reconsider k = (multcomplexcpfunc A).(c,f) as Element of PFuncs(A,COMPLEX);
  set j = (addcpfunc A).(g,h);
  dom g = dom f by Th7;
  then dom h /\ dom g = dom f /\ dom f by Th7;
  then
A1: dom j = dom f by Th4;
A2: now
    let x be Element of A;
    assume
A3: x in dom j;
    then x in dom(b(#)f) by A1,VALUED_1:def 5;
    then (b(#)f).x = b*f.x by VALUED_1:def 5;
    then
A4: h.x = b*f.x by Def4;
    x in dom(a(#)f) by A1,A3,VALUED_1:def 5;
    then (a(#)f).x = a*f.x by VALUED_1:def 5;
    then g.x = a*f.x by Def4;
    then g.x + h.x = (a+b)*(f.x) by A4;
    hence j.x = (a+b)*(f.x) by A3,Th4
      .= k.x by A1,A3,Th7;
  end;
  dom k = dom f by Th7;
  hence thesis by A1,A2,PARTFUN1:5;
end;
