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

theorem
  G1 == G2 iff G1.allSpanningSG() = G2.allSpanningSG()
proof
  hereby
    assume A1: G1 == G2;
    then the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 34;
    then G1 is spanning Subgraph of G2 & G2 is spanning Subgraph of G1
      by A1, GLIB_000:87, GLIB_000:def 33;
    then G1.allSpanningSG() c= G2.allSpanningSG() &
      G2.allSpanningSG() c= G1.allSpanningSG() by Th65;
    hence G1.allSpanningSG() = G2.allSpanningSG()
      by XBOOLE_0:def 10;
  end;
  assume G1.allSpanningSG() = G2.allSpanningSG();
  then G1 is spanning Subgraph of G2 & G2 is spanning Subgraph of G1
    by Th65;
  hence thesis by GLIB_000:87;
end;
