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 holds TotFuncs f c= Funcs(X,Y)
proof
  let f be PartFunc of X,Y;
  per cases;
  suppose Y = {} & X <> {};
    hence thesis;
  end;
  suppose
A1: Y <> {} or X = {};
    let g be object;
    assume g in TotFuncs f;
    then g is Function of X,Y by Th81;
    hence thesis by A1,Th8;
  end;
end;
