reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);

theorem Th15:
  (addpfunc A).(RealPFuncZero A,f) = f
proof
  set h = (addpfunc A).(RealPFuncZero A,f);
  dom h = dom(RealPFuncZero A) /\ dom f by Th6;
  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: (RealPFuncZero A).x = 0 by FUNCOP_1:7;
    assume x in dom f;
    hence h.x=0+ f.x by A1,A2,Th6
      .= f.x;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
