
theorem Th55:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TA-structure for t being type of T, a
being adjective of T for A,B being Subset of the adjectives of T st B = A \/ {a
  } & B is_applicable_to t holds a ast (A ast t) = B ast t
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  let t be type of T, a be adjective of T;
  let A,B be Subset of the adjectives of T such that
A1: B = A \/ {a} and
A2: B is_applicable_to t;
A3: A is_applicable_to t by A1,A2,Th54,XBOOLE_1:7;
A4: {a} c= B by A1,XBOOLE_1:7;
A5: A c= B by A1,XBOOLE_1:7;
  types a /\ downarrow (A ast t) = types B /\ downarrow t
  proof
    thus types a /\ downarrow (A ast t) c= types B /\ downarrow t
    proof
      let x be object;
      assume
A6:   x in types a /\ downarrow (A ast t);
      then reconsider t9 = x as type of T;
      x in types a by A6,XBOOLE_0:def 4;
      then a in adjs t9 by Th13;
      then
A7:   {a} c= adjs t9 by ZFMISC_1:31;
      x in downarrow (A ast t) by A6,XBOOLE_0:def 4;
      then
A8:   t9 <= A ast t by WAYBEL_0:17;
      then
A9:   adjs (A ast t) c= adjs t9 by Th10;
      A ast t <= t by A3,Th49;
      then t9 <= t by A8,YELLOW_0:def 2;
      then
A10:  t9 in downarrow t by WAYBEL_0:17;
      A c= adjs (A ast t) by A3,Th50;
      then A c= adjs t9 by A9;
      then B c= adjs t9 by A1,A7,XBOOLE_1:8;
      then t9 in types B by Th14;
      hence thesis by A10,XBOOLE_0:def 4;
    end;
    let x be object;
    assume
A11: x in types B /\ downarrow t;
    then reconsider t9 = x as type of T;
    x in downarrow t by A11,XBOOLE_0:def 4;
    then
A12: t9 <= t by WAYBEL_0:17;
    x in types B by A11,XBOOLE_0:def 4;
    then
A13: B c= adjs t9 by Th14;
    then A c= adjs t9 by A5;
    then t9 <= A ast t by A12,Th52;
    then
A14: t9 in downarrow (A ast t) by WAYBEL_0:17;
    a in B by A4,ZFMISC_1:31;
    then t9 in types a by A13,Th13;
    hence thesis by A14,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
