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
  holds G1 == G2 iff e in the_Edges_of G2
proof
  let G2;
  let v1, v2 be Vertex of G2;
  let e be object;
  let G1 be addEdge of G2,v1,e,v2;
  hereby
    assume G1 == G2;
    then A1: the_Edges_of G1 = the_Edges_of G2 by GLIB_000:def 34;
    per cases;
    suppose e in the_Edges_of G2;
      hence e in the_Edges_of G2;
    end;
    suppose not e in the_Edges_of G2;
      then the_Edges_of G2 \/ {e} c= the_Edges_of G2 by A1, Def11;
      hence e in the_Edges_of G2 by ZFMISC_1:39;
    end;
  end;
  thus thesis by Def11;
end;
