 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 Th9:
  CPFuncZero A is_a_unity_wrt addcpfunc A
proof
  thus CPFuncZero A is_a_left_unity_wrt addcpfunc A
  proof
    let f;
    set h = (addcpfunc A).(CPFuncZero A,f);
    dom h = dom(CPFuncZero A) /\ dom f by Th4;
    then dom h = A /\ dom f by FUNCOP_1:13;
    then
A1: dom h = dom f by XBOOLE_1:28;
    now
      let x be Element of A;
A2:   (CPFuncZero A).x = 0c by FUNCOP_1:7;
      assume x in dom f;
      hence h.x=0c+ f.x by A1,A2,Th4
      .= f.x;
    end;
    hence thesis by A1,PARTFUN1:5;
  end;
  let f;
  set h = (addcpfunc A).(f,CPFuncZero A);
  dom h = dom(CPFuncZero A) /\ dom f by Th4;
  then dom h = A /\ dom f by FUNCOP_1:13;
  then
A3: dom h = dom f by XBOOLE_1:28;
    now
      let x be Element of A;
A4:   (CPFuncZero A).x = 0c by FUNCOP_1:7;
      assume x in dom f;
      hence h.x=0c+ f.x by A3,A4,Th4
      .= f.x;
    end;
    hence thesis by A3,PARTFUN1:5;
  end;
