
theorem Th100:
  for G being _Graph, u,v,w being Vertex of G st v is endvertex & u <> w
  holds not u,v are_adjacent or not v,w are_adjacent
proof
  let G be _Graph, u,v,w being Vertex of G;
  assume A1: v is endvertex & u <> w;
  assume A2: u,v are_adjacent;
  consider e being object such that
    A3: v.edgesInOut() = {e} & not e Joins v,v,G by A1, GLIB_000:def 51;
  e in v.edgesInOut() by A3, TARSKI:def 1;
  then consider v9 being Vertex of G such that
    A4: e Joins v,v9,G by GLIB_000:64;
  consider e8 being object such that
    A5: e8 Joins v,u,G by A2, CHORD:def 3;
  e8 is set by TARSKI:1;
  then e8 in v.edgesInOut() by A5, GLIB_000:64;
  then e8 = e by A3, TARSKI:def 1;
  then A6: v = v & v9 = u or v = u & v9 = v by A4, A5, GLIB_000:15;
  not ex e9 being object st e9 Joins v,w,G
  proof
    given e9 being object such that
      A7: e9 Joins v,w,G;
    e9 is set by TARSKI:1;
    then e9 in v.edgesInOut() by A7, GLIB_000:64;
    then e9 = e by A3, TARSKI:def 1;
    then v = v & w = v9 or v = v9 & w = v by A4, A7, GLIB_000:15;
    hence contradiction by A1, A6;
  end;
  hence not v,w are_adjacent by CHORD:def 3;
end;
