
theorem Th31:
  for G1 being non _trivial _finite Tree-like _Graph
  for G2 being non spanning connected Subgraph of G1
  ex v being Vertex of G1 st v is endvertex & not v in the_Vertices_of G2
proof
  let G1 be non _trivial _finite Tree-like _Graph;
  let G2 be non spanning connected Subgraph of G1;
  assume A1: for v being Vertex of G1 holds not v is endvertex or
    v in the_Vertices_of G2;
  A2: not the_Vertices_of G1 c= the_Vertices_of G2
    by GLIB_000:def 33, XBOOLE_0:def 10;
  :: Now we show the negation
  for x being object st x in the_Vertices_of G1 holds x in the_Vertices_of G2
  proof
    let x be object;
    assume x in the_Vertices_of G1;
    then reconsider v = x as Vertex of G1;
    consider v1, v2 being Vertex of G1 such that
      A3: v1 <> v2 & v1 is endvertex & v2 is endvertex and
      A4: v in G1.pathBetween(v1,v2).vertices() by Th30;
    reconsider w1 = v1, w2 = v2 as Vertex of G2 by A1, A3;
    v in G2.pathBetween(w1,w2).vertices() by A4, HELLY:33, GLIB_001:98;
    hence thesis;
  end;
  hence contradiction by A2, TARSKI:def 3;
end;
