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

theorem Th140:
  for G2 for v1, e, v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  holds G2 is inducedSubgraph of G1, the_Vertices_of G2
proof
  let G2;
  let v1,e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  per cases;
  suppose v1 in the_Vertices_of G2 & not v2 in the_Vertices_of G2 &
    not e in the_Edges_of G2;
    then the_Vertices_of G1 = the_Vertices_of G2 \/ {v2} by Def12;
    then A1: the_Vertices_of G2 is non empty Subset of the_Vertices_of G1
      by XBOOLE_1:7;
    A2: G2 is Subgraph of G1 by Th61;
    the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G2) by Th139;
    hence thesis by A1, A2, GLIB_000:def 37;
  end;
  suppose not v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
    not e in the_Edges_of G2;
    then the_Vertices_of G1 = the_Vertices_of G2 \/ {v1} by Def12;
    then A3: the_Vertices_of G2 is non empty Subset of the_Vertices_of G1
      by XBOOLE_1:7;
    A4: G2 is Subgraph of G1 by Th61;
    the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G2) by Th139;
    hence thesis by A3, A4, GLIB_000:def 37;
  end;
  suppose not (v1 in the_Vertices_of G2 & not v2 in the_Vertices_of G2 &
      not e in the_Edges_of G2) & not (
    not v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
    not e in the_Edges_of G2);
    then A5: G1 == G2 by Def12;
    then A6: the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 34;
    G1 is inducedSubgraph of G1, the_Vertices_of G1 by GLIB_000:100;
    hence thesis by A5, A6, GLIB_000:101;
  end;
end;
