
theorem
  for I being non empty set for A being ManySortedSet of I for i being
Element of I for S being non trivial set holds A+*(i,S) is non trivial-yielding
proof
  let I be non empty set;
  let A be ManySortedSet of I;
  let i be Element of I;
  let S be non trivial set;
  take a = A+*(i,S).i;
  dom A = I by PARTFUN1:def 2;
  then i in dom A;
  then i in dom (A+*(i,S)) by FUNCT_7:30;
  hence a in rng (A+*(i,S)) by FUNCT_1:def 3;
  dom A = I by PARTFUN1:def 2;
  hence thesis by FUNCT_7:31;
end;
