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.order() = 1 iff ex v being Vertex of
  G st the_Vertices_of G = {v}
proof
  let G be _Graph;
  hereby
    assume G.order() = 1;
    then consider v being object such that
A1: the_Vertices_of G = {v} by CARD_2:42;
    reconsider v as Vertex of G by A1,TARSKI:def 1;
    take v;
    thus the_Vertices_of G = {v} by A1;
  end;
  given v being Vertex of G such that
A2: the_Vertices_of G = {v};
  thus thesis by A2,CARD_1:30;
end;
