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

theorem
  G1 == G2 iff G1.allSG() = G2.allSG()
proof
  hereby
    assume G1 == G2;
    then G1 is Subgraph of G2 & G2 is Subgraph of G1 by GLIB_000:87;
    then G2.allSG() c= G1.allSG() &
      G1.allSG() c= G2.allSG() by Th29;
    hence G1.allSG() = G2.allSG() by XBOOLE_0:def 10;
  end;
  assume G1.allSG() = G2.allSG();
  then G1 is Subgraph of G2 & G2 is Subgraph of G1 by Th29;
  hence G1 == G2 by GLIB_000:87;
end;
