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

theorem Th93:
  for G2, V for G1 being addVertices of G2, V
  st V \ the_Vertices_of G2 <> {}
  holds G1 is non _trivial non connected non Tree-like non complete
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  assume V \ the_Vertices_of G2 <> {};
  then consider v1 being object such that
    A2: v1 in V \ the_Vertices_of G2 by XBOOLE_0:def 1;
  A3: v1 in V & not v1 in the_Vertices_of G2 by A2, XBOOLE_0:def 5;
  then v1 in the_Vertices_of G2 \/ V by XBOOLE_0:def 3;
  then reconsider v1 as Vertex of G1 by Def10;
  set v2 = the Vertex of G2;
  v1 <> v2 by A3;
  then A5: card {v1, v2} = 2 by CARD_2:57;
  v2 in the_Vertices_of G2 \/ V by XBOOLE_0:def 3;
  then v2 in the_Vertices_of G1 by Def10;
  then A6: 2 c= card the_Vertices_of G1 by A5, CARD_1:11, ZFMISC_1:32;
  card the_Vertices_of G1 <> 1
  proof
    assume card the_Vertices_of G1 = 1;
    then 1 in 1 by A6, CARD_2:69;
    hence contradiction;
  end;
  hence A7: G1 is non _trivial by GLIB_000:def 19;
  v1 is isolated by A2, Th92;
  hence G1 is non connected non Tree-like non complete by A7, GLIB_002:2;
end;
