
theorem Th57:
  for T being non empty non void reflexive transitive
TAS-structure for t being type of T, v being FinSequence of the adjectives of T
  st v is_properly_applicable_to t holds v is_applicable_to t
proof
  let T be non empty non void reflexive transitive TAS-structure;
  let t be type of T;
  let v be FinSequence of the adjectives of T;
  assume
A1: for i being Nat, a being adjective of T, s being type of
T st i in dom v & a = v.i & s = apply(v,t).i holds a is_properly_applicable_to
  s;
  let i be Nat, a be adjective of T, s be type of T such that
A2: i in dom v and
A3: a = v.i and
A4: s = apply(v, t).i;
  a is_properly_applicable_to s by A1,A2,A3,A4;
  hence thesis;
end;
