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

theorem Th116:
  for G2 for v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2 st G2 is loopless & v1 <> v2
  holds G1 is loopless
proof
  let G2;
  let v1, v2 be Vertex of G2;
  let e be object;
  let G1 be addEdge of G2,v1,e,v2;
  assume that
    A1: G2 is loopless and
    A2: v1 <> v2;
  per cases;
  suppose e in the_Edges_of G2;
    then G1 == G2 by Def11;
    hence thesis by A1, GLIB_000:89;
  end;
  suppose A3: not e in the_Edges_of G2;
    not ex e1 being object st e1 in the_Edges_of G1 &
      (the_Source_of G1).e1 = (the_Target_of G1).e1
    proof
      given e1 being object such that
        A4: e1 in the_Edges_of G1 and
        A5: (the_Source_of G1).e1 = (the_Target_of G1).e1;
      per cases;
      suppose A6: e1 in the_Edges_of G2;
        reconsider e1 as set by TARSKI:1;
        (the_Source_of G1).e1 = (the_Source_of G2).e1 &
          (the_Target_of G1).e1 = (the_Target_of G2).e1 by A6, Def9;
        hence contradiction by A1, A5, A6, GLIB_000:def 18;
      end;
      suppose A7: not e1 in the_Edges_of G2;
        the_Edges_of G1 = the_Edges_of G2 \/ {e} by A3, Def11;
        then e1 in {e} by A4, A7, XBOOLE_0:def 3;
        then e1 = e by TARSKI:def 1;
        then e1 DJoins v1,v2,G1 by A7, Th109;
        then (the_Source_of G1).e1 = v1 & (the_Target_of G1).e1 = v2
          by GLIB_000:def 14;
        hence contradiction by A2, A5;
      end;
    end;
    hence thesis by GLIB_000:def 18;
  end;
end;
