
theorem
  for T being non empty TA-structure for a being adjective of T holds
  types a = {t where t is type of T: a in adjs t}
proof
  let T be non empty TA-structure;
  let a be adjective of T;
  set X = {t where t is type of T: a in adjs t};
  X c= the carrier of T
  proof
    let x be object;
    assume x in X;
    then ex t being type of T st x = t & a in adjs t;
    hence thesis;
  end;
  then reconsider X as Subset of T;
  for x being object
holds x in X iff ex t being type of T st x = t & a in adjs t;
  hence thesis by Def12;
end;
