
theorem Th33:
  for T being _Tree for a, b being Vertex of T for t being
  _Subtree of T for a9, b9 being Vertex of t st a = a9 & b = b9 holds T
  .pathBetween(a,b) = t.pathBetween(a9,b9)
proof
  let T be _Tree;
  let a, b be Vertex of T;
  let t be _Subtree of T;
  let a9, b9 be Vertex of t such that
A1: a = a9 and
A2: b = b9;
  set tp = t.pathBetween(a9,b9);
  reconsider tp9 = tp as Walk of T by GLIB_001:167;
A3: tp is_Walk_from a9, b9 by Def2;
A4: tp9.last() = tp.last() .= b by A2,A3;
  tp9.first() = tp.first() .= a by A1,A3;
  then tp9 is Path-like & tp9 is_Walk_from a, b by A4,GLIB_001:176;
  hence thesis by Def2;
end;
