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

theorem Th48:
  G2.allInducedSG() c= G1.allInducedSG() iff
    ex V being non empty Subset of the_Vertices_of G1
    st G2 is inducedSubgraph of G1,V
proof
  hereby
    assume A1: G2.allInducedSG() c= G1.allInducedSG();
    G2 | _GraphSelectors in G2.allInducedSG() by Th47;
    then consider V being non empty Subset of the_Vertices_of G1 such that
      A2: G2 | _GraphSelectors is plain inducedSubgraph of G1,V by A1, Th45;
    take V;
    G2 == G2 | _GraphSelectors by GLIB_000:128;
    hence G2 is inducedSubgraph of G1, V by A2, GLIB_000:101;
  end;
  given V being non empty Subset of the_Vertices_of G1 such that
    A3: G2 is inducedSubgraph of G1,V;
  now
    let x be object;
    assume x in G2.allInducedSG();
    then consider V2 being non empty Subset of the_Vertices_of G2 such that
      A4: x is plain inducedSubgraph of G2,V2 by Th45;
    the_Vertices_of G2 = V by A3, GLIB_000:def 37;
    then A5: V2 c= V;
    then A6: V2 is non empty Subset of the_Vertices_of G1 by XBOOLE_1:1;
    then x is inducedSubgraph of G1,V2 by A3, A4, A5, CHORD:29;
    hence x in G1.allInducedSG() by A4, A6, Th45;
  end;
  hence thesis by TARSKI:def 3;
end;
