reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th16:
  for D be non empty set holds [#](D) --> (0 qua Real)
  is_a_unity_wrt addpfunc(D)
proof
  let D be non empty set;
  set F = [#](D) --> In(0,REAL);
A1: dom F = D by FUNCOP_1:13;
A2: now
    let G be Element of PFuncs(D,REAL);
A3: now
      let d be Element of D;
      assume d in dom(G+F);
      hence (G+F).d = G.d+F.d by VALUED_1:def 1
        .= G.d + 0 by FUNCOP_1:7
        .= G.d;
    end;
    dom G /\ D = dom G by XBOOLE_1:28;
    then dom(G+F) = dom G by A1,VALUED_1:def 1;
    then G+F = G by A3,PARTFUN1:5;
    hence addpfunc(D).(G,F) = G by Def4;
  end;
  addpfunc(D) is commutative by Th14;
  hence thesis by A2,BINOP_1:5;
end;
