
theorem
  for T being adj-structured with_suprema antisymmetric commutative non
empty non void reflexive transitive Noetherian TAS-structure for t being type
  of T, a being adjective of T st a is_properly_applicable_to t holds radix (a
  ast t) = radix t
proof
  let T be adj-structured with_suprema antisymmetric commutative non empty
  non void reflexive transitive Noetherian TAS-structure;
  let t be type of T, a be adjective of T;
A1: a in adjs t or not a in adjs t;
  assume a is_properly_applicable_to t;
  then a ast t = t or [a ast t, t] in T@--> by A1,Def31,Th24;
  then T@--> reduces a ast t,t by REWRITE1:12,15;
  then
A2: a ast t, t are_convertible_wrt T@--> by REWRITE1:25;
  T@--> is with_Church-Rosser_property with_UN_property
  strongly-normalizing Relation by Th69,Th77;
  hence thesis by A2,REWRITE1:55;
end;
