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

theorem
  (multpfunc A).(RealPFuncUnit A,f) = f
proof
  set h = (multpfunc A).(RealPFuncUnit A,f);
  dom h = dom(RealPFuncUnit A) /\ dom f by Th7;
  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;
    assume x in dom f;
    then h.x = (RealPFuncUnit A).x * f.x by A1,Th7;
    then h.x = 1 * f.x by FUNCOP_1:7;
    hence h.x = f.x;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
