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

theorem Th49:
  G1 == G2 iff G1.allInducedSG() = G2.allInducedSG()
proof
  set V1 = the_Vertices_of G1, V2 = the_Vertices_of G2;
  hereby
    assume A1: G1 == G2;
    G1 is inducedSubgraph of G1, V1 & G2 is inducedSubgraph of G2, V2
      by GLIB_000:100;
    then A2: G1 is inducedSubgraph of G2, V2 & G2 is inducedSubgraph of G1, V1
      by A1, GLIB_000:101;
    V1 c= V1 & V2 c= V2;
    then G1.allInducedSG() c= G2.allInducedSG() &
      G2.allInducedSG() c= G1.allInducedSG() by A2, Th48;
    hence G1.allInducedSG() = G2.allInducedSG()
      by XBOOLE_0:def 10;
  end;
  assume A3: G1.allInducedSG() = G2.allInducedSG();
  then consider V2 being non empty Subset of the_Vertices_of G2 such that
    A4: G1 is inducedSubgraph of G2, V2 by Th48;
  consider V1 being non empty Subset of the_Vertices_of G1 such that
    A5: G2 is inducedSubgraph of G1, V1 by A3, Th48;
  thus thesis by A4, A5, GLIB_000:87;
end;
