reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem
  for f be Function, d,e,i be set holds f+*(i,d)+*(i,e) = f+*(i,e)
proof
  let f be Function, d,e,i be set;
A1: dom(i.-->d) = {i}
    .= dom(i.-->e);
  per cases;
  suppose
A2: i in dom f;
    then i in dom(f+*(i,d)) by Th29;
    hence f+*(i,d)+*(i,e) = f+*(i,d)+*(i.-->e) by Def2
      .= f+*(i.-->d)+*(i.-->e) by A2,Def2
      .= f+*((i.-->d)+*(i.-->e)) by FUNCT_4:14
      .= f+*(i.-->e) by A1,FUNCT_4:19
      .= f+*(i,e) by A2,Def2;
  end;
  suppose
    not i in dom f;
    hence thesis by Def2;
  end;
end;
