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
  for I being non empty set
  for i being Element of I
  for A being ManySortedSet of I
  for x st x in A.i holds i-singleton x is ManySortedSubset of A
  proof
    let I be non empty set;
    let i be Element of I;
    let A be ManySortedSet of I;
    let x;
    assume A1: x in A.i;
    let y be object;
    assume y in I; then
    (y = i implies (i-singleton x).y = {x}) &
    (y <> i implies (i-singleton x).y = {}) by Th6;
    hence (i-singleton x).y c= A.y by A1,ZFMISC_1:31;
  end;
