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

theorem
  for G2 for v1,v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2
  st not e in the_Edges_of G2 & v1,v2 are_adjacent
  holds G1 is non non-multi
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: not e in the_Edges_of G2 and
    A2: v1,v2 are_adjacent;
  ex e1,e2,u1,u2 being object st e1 Joins u1,u2,G1 & e2 Joins u1,u2,G1
    & e1 <> e2
  proof
    consider e1 being object such that
      A3: e1 Joins v1,v2,G2 by A2, CHORD:def 3;
    take e1,e,v1,v2;
    thus e1 Joins v1,v2,G1 by A3, Th74;
    e DJoins v1,v2,G1 by A1, Th109;
    hence e Joins v1,v2,G1 by GLIB_000:16;
    thus e1 <> e by A1, A3, GLIB_000:def 13;
  end;
  hence thesis by GLIB_000:def 20;
end;
