
theorem Th4:
  for a1,a2 being set st a1 <> a2 for A being AdjectiveStr st the
adjectives of A = {a1,a2} & (the non-op of A).a1 = a2 & (the non-op of A).a2 =
  a1 holds A is non void involutive without_fixpoints
proof
  let a1,a2 be set such that
A1: a1 <> a2;
  let A be AdjectiveStr such that
A2: the adjectives of A = {a1,a2} and
A3: (the non-op of A).a1 = a2 and
A4: (the non-op of A).a2 = a1;
  thus the adjectives of A is non empty by A2;
  hereby
    let a be adjective of A;
    a = a1 or a = a2 by A2,TARSKI:def 2;
    hence non-non-a = a by A3,A4;
  end;
  let a be adjective of A;
  assume
A5: non-a = a;
  a = a1 or a = a2 by A2,TARSKI:def 2;
  hence thesis by A1,A3,A4,A5;
end;
