 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 Th10:
  (addcpfunc A).(f,(multcomplexcpfunc A).(-1r,f)) = (CPFuncZero A)|(dom f)
proof
  reconsider g = (multcomplexcpfunc A).(R,f) as Element of PFuncs(A,COMPLEX);
  set h = (addcpfunc A).(f,g);
  dom (CPFuncZero A) = A by FUNCOP_1:13;
  then dom ((CPFuncZero A)|(dom f)) = A /\ dom f by RELAT_1:61;
  then
A1: dom ((CPFuncZero A)|(dom f)) = dom f by XBOOLE_1:28;
A2: dom h = dom g /\ dom f by Th4
    .= dom f /\ dom f by Th7;
  now
    let x be Element of A;
    assume
A3: x in dom f;
    then
A4: x in dom((-1r)(#)f) by VALUED_1:def 5;
    thus h.x = f.x + g.x by A2,A3,Th4
      .= f.x + ((-1r)(#)f).x by Def4
      .= f.x + (-1r) * f.x by A4,VALUED_1:def 5
      .= (CPFuncZero A).x by FUNCOP_1:7,COMPLEX1:def 4
      .= ((CPFuncZero A)|(dom f)).x by A3,FUNCT_1:49;
  end;
  hence thesis by A1,A2,PARTFUN1:5;
end;
