reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;

theorem
  for f being PartFunc of X,{y} for g being Function of X,{y} holds
  TotFuncs f = {g}
proof
  let f be PartFunc of X,{y};
  let g be Function of X,{y};
  for h being object holds h in TotFuncs f iff h = g
  proof
    let h be object;
    thus h in TotFuncs f implies h = g
    proof
      assume h in TotFuncs f;
      then h is Function of X,{y} by Th81;
      hence thesis by Th50;
    end;
    f tolerates g by PARTFUN1:61;
    hence thesis by PARTFUN1:def 5;
  end;
  hence thesis by TARSKI:def 1;
end;
