reserve X for set;
reserve UN for Universe;

theorem Th33:
  for I,u,v being Element of UN
  for x being UN-valued ManySortedSet of I st
  I = { {}, {{}} } & x.{} = u & x.{{}} = v holds
  disjoint-union x = [: u, { {} } :] \/ [: v, { {{}} }:]
  proof
    let I,u,v be Element of UN;
    let x be UN-valued ManySortedSet of I;
    assume
A1: I = { {}, {{}} } & x.{} = u & x.{{}} = v;
    {}UN is Element of UN & {{}UN} is Element of UN;
    hence thesis by A1,Th32;
  end;
