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

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

assume G1 == G2 ; :: thesis: G1 .allTrees() = G2 .allTrees()
then G2 is removeLoops of G1 by GLIB_009:58;
hence G1 .allTrees() = G2 .allTrees() by Th146; :: thesis: verum

end;

assume G1 .allTrees() = G2 .allTrees() ; :: thesis: G1 == G2
then ( G1 is Subgraph of G2 & G2 is Subgraph of G1 ) by Th145;
hence G1 == G2 by GLIB_000:87; :: thesis: verum