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 G is _trivial iff the_Vertices_of G is trivial
proof
  let G be _Graph;
  thus G is _trivial implies the_Vertices_of G is trivial;
  assume A1: the_Vertices_of G is trivial;
  assume G is non _trivial;
  then consider v1, v2 being Vertex of G such that
    A2: v1 <> v2 by Th21;
  thus contradiction by A1, A2, ZFMISC_1:def 10;
end;
