reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem
  for G being _Graph
  holds (ex v being Vertex of G st v is endvertex) implies G is non _trivial
proof
  let G be _Graph;
  given v being Vertex of G such that
    A1: v is endvertex;
  set G2 = the removeVertex of G, v;
  consider e being object such that
    A2: v.edgesInOut() = {e} & not e Joins v,v,G by A1;
  set w = v.adj(e);
  A3: e in v.edgesInOut() by A2, TARSKI:def 1;
  for u being Vertex of G holds the_Vertices_of G <> {u}
  proof
    let u be Vertex of G;
    assume the_Vertices_of G = {u};
    then v = u & w = u by TARSKI:def 1;
    hence contradiction by A2, A3, Th67;
  end;
  hence G is non _trivial by Th22;
end;
