theorem Th81:
  Subformulae x 'in' y = { x 'in' y }
proof
  now
    let a be object;
    thus a in Subformulae x 'in' y implies a in { x 'in' y }
    proof
      assume a in Subformulae x 'in' y;
      then consider F such that
A1:   F = a and
A2:   F is_subformula_of x 'in' y by Def42;
      F = x 'in' y by A2,Th77;
      hence thesis by A1,TARSKI:def 1;
    end;
    assume a in { x 'in' y };
    then
A3: a = x 'in' y by TARSKI:def 1;
    x 'in' y is_subformula_of x 'in' y by Th59;
    hence a in Subformulae x 'in' y by A3,Def42;
  end;
  hence thesis by TARSKI:2;
end;
