reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th117:
  for G2 for v being Vertex of G2, e being object
  for G1 being addEdge of G2,v,e,v
  st G2 is non loopless or not e in the_Edges_of G2
  holds G1 is non loopless
proof
  let G2;
  let v be Vertex of G2,e be object;
  let G1 be addEdge of G2,v,e,v;
  assume G2 is non loopless or not e in the_Edges_of G2;
  then per cases;
  suppose G2 is non loopless;
    hence G1 is non loopless;
  end;
  suppose not e in the_Edges_of G2;
    then e DJoins v,v,G1 by Th109;
    then e in the_Edges_of G1 & (the_Source_of G1).e = v &
      (the_Target_of G1).e = v by GLIB_000:def 14;
    hence thesis by GLIB_000:def 18;
  end;
end;
