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

theorem Th142:
  for G2 for v1 being set, e being object, v2 being Vertex of G2
  for G1 being addAdjVertex of G2,v1,e,v2
  st not e in the_Edges_of G2 & not v1 in the_Vertices_of G2
  holds G2 is removeVertex of G1, v1
proof
  let G2;
  let v1 be set, e be object, v2 be Vertex of G2;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2 & not v1 in the_Vertices_of G2;
  then the_Vertices_of G1 = the_Vertices_of G2 \/ {v1} by Def14;
  then the_Vertices_of G2 = the_Vertices_of G1 \ {v1} by A1, ZFMISC_1:117;
  hence thesis by Th140;
end;
