reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem
  for G2 being _Graph, v, V being set, G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds G2 is removeVertex of G1, v
proof
  let G2 be _Graph, v, V be set;
  let G1 be addAdjVertexAll of G2,v,V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  A2: G2 is Subgraph of G1 by GLIB_006:57;
  set V1 = the_Vertices_of G1 \ {v};
  the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by A1, Def4;
  then A3: the_Vertices_of G2 = V1 by A1, ZFMISC_1:117;
  then reconsider V1 as non empty Subset of the_Vertices_of G1;
  the_Edges_of G2 = G1.edgesBetween(V1) by A1, A3, Th53;
  hence thesis by A2, A3, GLIB_000:def 37;
end;
