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 a1,a2,a3,a4,a5,a6,a7,a8,a9 being object holds
  {a1}\/{a2,a3,a4,a5,a6,a7,a8,a9} = {a1,a2,a3,a4,a5,a6,a7,a8,a9}
  proof
    let a1,a2,a3,a4,a5,a6,a7,a8,a9 be object;
    thus {a1}\/{a2,a3,a4,a5,a6,a7,a8,a9} c= {a1,a2,a3,a4,a5,a6,a7,a8,a9}
    proof
      let x be object; assume x in {a1}\/{a2,a3,a4,a5,a6,a7,a8,a9}; then
      x in {a1} or x in {a2,a3,a4,a5,a6,a7,a8,a9} by XBOOLE_0:def 3; then
      x = a1 or x = a2 or x = a3 or x = a4 or x = a5 or x = a6 or x = a7 or
      x = a8 or x = a9 by TARSKI:def 1,ENUMSET1:def 6;
      hence thesis by ENUMSET1:def 7;
    end;
    let x be object; assume
    x in {a1,a2,a3,a4,a5,a6,a7,a8,a9}; then
    x = a1 or x = a2 or x = a3 or x = a4 or x = a5 or x = a6 or x = a7 or
    x = a8 or x = a9 by ENUMSET1:def 7; then
    x in {a1} or x in {a2,a3,a4,a5,a6,a7,a8,a9} by TARSKI:def 1,ENUMSET1:def 6;
    hence thesis by XBOOLE_0:def 3;
  end;
