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 _trivial loopless _Graph holds the_Edges_of G = {}
proof
  let G be _trivial loopless _Graph;
  consider v being Vertex of G such that
A1: the_Vertices_of G = {v} by Th22;
  now
    assume the_Edges_of G <> {};
    then consider e being object such that
A2: e in the_Edges_of G by XBOOLE_0:def 1;
    (the_Target_of G).e in {v} by A1,A2,FUNCT_2:5;
    then
A3: (the_Target_of G).e = v by TARSKI:def 1;
    (the_Source_of G).e in {v} by A1,A2,FUNCT_2:5;
    then (the_Source_of G).e = v by TARSKI:def 1;
    hence contradiction by A2,A3,Def18;
  end;
  hence thesis;
end;
