let G1, G2 be _Graph; :: thesis: ( G1 == G2 iff G1 .allSG() = G2 .allSG() )

hereby :: thesis: ( G1 .allSG() = G2 .allSG() implies G1 == G2 )

assume G1 == G2 ; :: thesis: G1 .allSG() = G2 .allSG()
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; :: thesis: verum

end;

assume G1 .allSG() = G2 .allSG() ; :: thesis: G1 == G2
then ( G1 is Subgraph of G2 & G2 is Subgraph of G1 ) by Th29;
hence G1 == G2 by GLIB_000:87; :: thesis: verum