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

theorem
  for G, G2 for v1,e,v2 being object, G1 being addAdjVertex of G,v1,e,v2
  st G1 == G2 holds G2 is addAdjVertex of G,v1,e,v2
proof
  let G, G2;
  let v1,e,v2 be object;
  let G1 be addAdjVertex of G,v1,e,v2;
  assume A1: G1 == G2;
  per cases;
  suppose A2: v1 in the_Vertices_of G & not v2 in the_Vertices_of G &
      not e in the_Edges_of G;
    then the_Vertices_of G1 = the_Vertices_of G \/ {v2} &
      the_Edges_of G1 = the_Edges_of G \/ {e} &
      the_Source_of G1 = the_Source_of G +* (e .--> v1) &
      the_Target_of G1 = the_Target_of G +* (e .--> v2) by Def12;
    then A3: the_Vertices_of G2 = the_Vertices_of G \/ {v2} &
      the_Edges_of G2 = the_Edges_of G \/ {e} &
      the_Source_of G2 = the_Source_of G +* (e .--> v1) &
      the_Target_of G2 = the_Target_of G +* (e .--> v2) by A1, GLIB_000:def 34;
    G2 is Supergraph of G1 by A1, Th62;
    then G2 is Supergraph of G by Th66;
    hence thesis by A2, A3, Def12;
  end;
  suppose A4: not v1 in the_Vertices_of G & v2 in the_Vertices_of G &
      not e in the_Edges_of G;
    then the_Vertices_of G1 = the_Vertices_of G \/ {v1} &
      the_Edges_of G1 = the_Edges_of G \/ {e} &
      the_Source_of G1 = the_Source_of G +* (e .--> v1) &
      the_Target_of G1 = the_Target_of G +* (e .--> v2) by Def12;
    then A5: the_Vertices_of G2 = the_Vertices_of G \/ {v1} &
      the_Edges_of G2 = the_Edges_of G \/ {e} &
      the_Source_of G2 = the_Source_of G +* (e .--> v1) &
      the_Target_of G2 = the_Target_of G +* (e .--> v2) by A1, GLIB_000:def 34;
    G2 is Supergraph of G1 by A1, Th62;
    then G2 is Supergraph of G by Th66;
    hence thesis by A4, A5, Def12;
  end;
  suppose A6: not (v1 in the_Vertices_of G & not v2 in the_Vertices_of G &
      not e in the_Edges_of G) & not (not v1 in the_Vertices_of G &
      v2 in the_Vertices_of G & not e in the_Edges_of G);
    then G == G1 by Def12;
    then A7: G == G2 by A1, GLIB_000:85;
    then G2 is Supergraph of G by Th62;
    hence thesis by A6, A7, Def12;
  end;
end;
