
theorem Th9:
  for T1,T2 being non empty TA-structure st the TA-structure of T1
  = the TA-structure of T2 holds T1 is adj-structured implies T2 is
  adj-structured
proof
  let T1,T2 be non empty TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2;
  assume the adj-map of T1 is join-preserving Function of T1, (BoolePoset the
  adjectives of T1) opp;
  then reconsider f = the adj-map of T1 as join-preserving Function of T1, (
  BoolePoset the adjectives of T1) opp;
  reconsider g = f as Function of T2, (BoolePoset the adjectives of T2) opp by
A1;
A2: the RelStr of T1 = the RelStr of T2 by A1;
  g is join-preserving
  proof
    let x,y be type of T2;
    reconsider x9 = x, y9 = y as type of T1 by A1;
    assume
A3: ex_sup_of {x,y}, T2;
    then
A4: ex_sup_of {x9,y9}, T1 by A2,YELLOW_0:14;
A5: f preserves_sup_of {x9,y9} by WAYBEL_0:def 35;
    hence ex_sup_of g.:{x,y}, (BoolePoset the adjectives of T2) opp by A1,A4;
    sup (f.:{x9,y9}) = f.sup {x9,y9} by A4,A5;
    hence thesis by A1,A2,A3,YELLOW_0:26;
  end;
  hence the adj-map of T2 is join-preserving Function of T2, (BoolePoset the
  adjectives of T2) opp by A1;
end;
