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

theorem Th34:
  for f be Function, i be set holds f+*(i,f.i) = f
proof
  let f be Function, i be set;
  per cases;
  suppose
A1: i in dom f;
    then
A2: i.-->f.i = f|{i} by Th6;
    thus f+*(i,f.i) = f +*(i.-->f.i) by A1,Def2
      .= f by A2,FUNCT_4:75;
  end;
  suppose
    not i in dom f;
    hence thesis by Def2;
  end;
end;
