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
  the_Edges_of G = {} implies G is simple
proof
  assume
A1: the_Edges_of G = {};
  then
A2: G is loopless;
  for e1,e2,v1,v2 being object holds e1 Joins v1,v2,G & e2 Joins v1,v2,G
  implies e1 = e2 by A1;
  then G is non-multi;
  hence thesis by A2;
end;
