
theorem
  for T being non empty non void reflexive transitive TA-structure for t
  being type of T, a being adjective of T holds a is_applicable_to t iff <*a*>
  is_applicable_to t
proof
  let T be non empty non void reflexive transitive TA-structure;
  let t be type of T;
  let a be adjective of T;
  set v = <*a*>;
A1: v.1 = a;
  hereby
    assume
A2: a is_applicable_to t;
    thus <*a*> is_applicable_to t
    proof
      let i be Nat, b be adjective of T, s be type of T;
      assume i in dom v;
      then i in Seg 1 by FINSEQ_1:38;
      then
A3:   i = 1 by FINSEQ_1:2,TARSKI:def 1;
      thus thesis by A2,A3,Def19;
    end;
  end;
  assume
A4: for i being Nat, a9 being adjective of T, s being type of
  T st i in dom v & a9 = v.i & s = apply(v,t).i holds a9 is_applicable_to s;
  len v = 1 by FINSEQ_1:40;
  then
A5: 1 in dom v by FINSEQ_3:25;
  apply(v,t).1 = t by Def19;
  hence thesis by A4,A5,A1;
end;
