
theorem Th64:
  for T being non empty non void reflexive transitive
TAS-structure for t being type of T, A being Subset of the adjectives of T st A
is_properly_applicable_to t ex B being Subset of the adjectives of T st B c= A
& B is_properly_applicable_to t & A ast t = B ast t & for C being Subset of the
  adjectives of T st C c= B & C is_properly_applicable_to t & A ast t = C ast t
  holds C = B
proof
  let T be non empty non void reflexive transitive TAS-structure;
  let t be type of T, A be Subset of the adjectives of T;
  defpred P[set] means ex B being Subset of the adjectives of T st $1 = B & $1
  c= A & B is_properly_applicable_to t & A ast t = B ast t;
  assume
A1: A is_properly_applicable_to t;
A2: ex a being finite set st P[a] by A1;
  consider B being finite set such that
A3: P[B] and
A4: for C being set st C c= B & P[C] holds C = B from MinimalFiniteSet(
  A2);
  reconsider B as Subset of the adjectives of T by A3;
  take B;
  thus B c= A & B is_properly_applicable_to t & A ast t = B ast t by A3;
  let C be Subset of the adjectives of T;
  assume
A5: C c= B;
  then C c= A by A3,XBOOLE_1:1;
  hence thesis by A4,A5;
end;
