
theorem
  for G being _Graph, v being Vertex of G st 4 c= G.order() & v is endvertex
  ex x,y,z being Vertex of G
  st v <> x & v <> y & v <> z & x <> y & x <> z & y <> z & v,x are_adjacent &
    not v,y are_adjacent & not v,z are_adjacent
proof
  let G be _Graph, v be Vertex of G;
  assume A1: 4 c= G.order() & v is endvertex;
  Segm 3 c= Segm 4 by NAT_1:39;
  then consider x,y being Vertex of G such that
    A2: x <> v & y <> v & x <> y & x,v are_adjacent & not v,y are_adjacent
    by A1, Th101, XBOOLE_1:1;
  consider z being object such that
    A3: z in the_Vertices_of G & v <> z & x <> z & y <> z by A1, Th21;
  reconsider z as Vertex of G by A3;
  take x,y,z;
  thus v <> x & v <> y & v <> z & x <> y & x <> z & y <> z by A2, A3;
  thus v,x are_adjacent by A2;
  thus not v,y are_adjacent by A2;
  thus not v,z are_adjacent by A1, A2, A3, Th100;
end;
