
theorem
  for T being non void TA-structure for t being type of T holds adjs t =
  {a where a is adjective of T: t in types a}
proof
  let T be non void TA-structure;
  let t be type of T;
  set X = {a where a is adjective of T: t in types a};
  thus adjs t c= X
  proof
    let x be object;
    assume
A1: x in adjs t;
    then reconsider a = x as adjective of T;
    t in types a by A1,Th13;
    hence thesis;
  end;
  let x be object;
  assume x in X;
  then ex a being adjective of T st x = a & t in types a;
  hence thesis by Th13;
end;
