
theorem Th101:
  for G being _Graph, v being Vertex of G st 3 c= G.order() & v is endvertex
  ex u,w being Vertex of G
  st u <> v & w <> v & u <> w & u,v are_adjacent & not v,w are_adjacent
proof
  let G be _Graph, v be Vertex of G;
  assume A1: 3 c= G.order() & v is endvertex;
  then A2: 3 c= card the_Vertices_of G;
  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 u being Vertex of G such that
    A4: e Joins v,u,G by GLIB_000:64;
  consider w being object such that
    A5: w in the_Vertices_of G & w <> v & w <> u by A2, PENCIL_1:6;
  reconsider w as Vertex of G by A5;
  take u,w;
  thus u <> v & w <> v & u <> w by A3, A4, A5;
  thus u,v are_adjacent by A4, CHORD:def 3;
  hence not v,w are_adjacent by A1, A5, Th100;
end;
