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

theorem
  G1 == G2 iff G1.allConnectedSG() = G2.allConnectedSG()
proof
  hereby
    assume G1 == G2;
    then G1 is Subgraph of G2 & G2 is Subgraph of G1 by GLIB_000:87;
    then G2.allConnectedSG() c= G1.allConnectedSG() &
      G1.allConnectedSG() c= G2.allConnectedSG() by Th129;
    hence G1.allConnectedSG() = G2.allConnectedSG() by XBOOLE_0:def 10;
  end;
  assume G1.allConnectedSG() = G2.allConnectedSG();
  then G2 is Subgraph of G1 & G1 is Subgraph of G2 by Th130;
  hence thesis by GLIB_000:87;
end;
