reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th45:
  G2 in G1.allInducedSG() iff
    ex V being non empty Subset of the_Vertices_of G1
    st G2 is plain inducedSubgraph of G1,V
proof
  hereby
    assume G2 in G1.allInducedSG();
    then consider V being non empty Subset of the_Vertices_of G1 such that
      A1: G2 = the plain inducedSubgraph of G1,V;
    take V;
    thus G2 is plain inducedSubgraph of G1,V by A1;
  end;
  given V being non empty Subset of the_Vertices_of G1 such that
    A2: G2 is plain inducedSubgraph of G1,V;
  set H = the plain inducedSubgraph of G1,V;
  G2 = H by A2, GLIB_000:93, GLIB_009:44;
  hence G2 in G1.allInducedSG();
end;
