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

theorem Th16:
  (addpfunc A).(f,(multrealpfunc A).(-1,f)) = (RealPFuncZero A)|( dom f)
proof
  reconsider g = (multrealpfunc A).(-jj,f) as Element of PFuncs(A,REAL);
  set h = (addpfunc A).(f,g);
  dom (RealPFuncZero A) = A by FUNCOP_1:13;
  then dom ((RealPFuncZero A)|(dom f)) = A /\ dom f by RELAT_1:61;
  then
A1: dom ((RealPFuncZero A)|(dom f)) = dom f by XBOOLE_1:28;
A2: dom h = dom g /\ dom f by Th6
    .= dom f /\ dom f by Th9;
  now
    let x be Element of A;
    assume
A3: x in dom f;
    then
A4: x in dom((-1)(#)f) by VALUED_1:def 5;
    thus h.x = f.x + g.x by A2,A3,Th6
      .= f.x + ((-1)(#)f).x by Def4
      .= f.x + (-1) * f.x by A4,VALUED_1:def 5
      .= (RealPFuncZero A).x by FUNCOP_1:7
      .= ((RealPFuncZero A)|(dom f)).x by A3,FUNCT_1:49;
  end;
  hence thesis by A1,A2,PARTFUN1:5;
end;
