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

theorem Th141:
  for G2 for v1 being Vertex of G2, e being object, v2 being set
  for G1 being addAdjVertex of G2,v1,e,v2
  st not e in the_Edges_of G2 & not v2 in the_Vertices_of G2
  holds G2 is removeVertex of G1, v2
proof
  let G2;
  let v1 be Vertex of G2, e be object, v2 be set;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2 & not v2 in the_Vertices_of G2;
  then the_Vertices_of G1 = the_Vertices_of G2 \/ {v2} by Def13;
  then the_Vertices_of G2 = the_Vertices_of G1 \ {v2} by A1, ZFMISC_1:117;
  hence thesis by Th140;
end;
