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

theorem Th147:
  for G1, G2 being loopless _Graph
  holds G1 == G2 iff G1.allTrees() = G2.allTrees()
proof
  let G1, G2 be loopless _Graph;
  hereby
    assume G1 == G2;
    then G2 is removeLoops of G1 by GLIB_009:58;
    hence G1.allTrees() = G2.allTrees() by Th146;
  end;
  assume G1.allTrees() = G2.allTrees();
  then G1 is Subgraph of G2 & G2 is Subgraph of G1 by Th145;
  hence thesis by GLIB_000:87;
end;
