reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th6:
  for I being non empty set
  for i,j being Element of I
  for x holds (i-singleton x).i = {x} & (i <> j implies (i-singleton x).j = {})
  proof
    let I be non empty set;
    let i,j be Element of I;
    let x;
    dom(EmptyMS I) = I by PARTFUN1:def 2;
    hence (i-singleton x).i = {x} by FUNCT_7:31;
    assume i <> j;
    hence (i-singleton x).j = (EmptyMS I).j by FUNCT_7:32 .= {};
  end;
