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

theorem Th147:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  st not v2 in the_Vertices_of G2 & not e in the_Edges_of G2
  holds G1 is non _trivial
proof
  let G2;
  let v1 be Vertex of G2, e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not v2 in the_Vertices_of G2 & not e in the_Edges_of G2;
  then consider G3 being addVertex of G2,v2 such that
    A2: G1 is addEdge of G3,v1,e,v2 by Th129;
  {v2} \ the_Vertices_of G2 <> {} by A1, ZFMISC_1:59;
  then G3 is non _trivial by Th93;
  hence G1 is non _trivial by A2;
end;
