
theorem Th27:
  for G being non _trivial _finite Tree-like _Graph, v being Vertex of G
  st G.order() = 2 holds v is endvertex
proof
  let G be non _trivial _finite Tree-like _Graph, v be Vertex of G;
  assume G.order() = 2;
  then card the_Vertices_of G = 2 by GLIB_000:def 24;
  then consider v1, v2 being object such that
    A1: v1 <> v2 & the_Vertices_of G = {v1,v2} by CARD_2:60;
  consider w1,w2 being Vertex of G such that
    A2: w1 <> w2 & w1 is endvertex & w2 is endvertex by GLIB_002:45;
  (w1 = v1 or w1 = v2) & (w2 = v1 or w2 = v2) by A1, TARSKI:def 2;
  then per cases by A2;
  suppose w1 = v1 & w2 = v2;
    hence thesis by A1, A2, TARSKI:def 2;
  end;
  suppose w1= v2 & w2 = v1;
    hence thesis by A1, A2, TARSKI:def 2;
  end;
end;
