reserve X for set;
reserve UN for Universe;

theorem
  for I,u,v being Element of UN
  for x being UN-valued ManySortedSet of I st
  I = { {}, {{}} } & x.{} = {u} & x.{{}} = {v} & u <> v holds
    disjoint-union x = disjoint-union (u,v)
  proof
    let I,u,v be Element of UN;
    let x be UN-valued ManySortedSet of I;
    assume that
A1: I = { {}, {{}} } and
A2: x.{} = {u} and
A3: x.{{}} = {v} and
A4: u <> v;
    disjoint-union(u,v) = { [u, {} ] , [v, {{}} ] } &
      disjoint-union x = [: {u}, { {} } :] \/ [: {v}, { {{}} }:]
      by A1,A2,A3,Th33,A4,FUNCT_4:67;
    hence thesis by Th31;
  end;
