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

theorem Th172:
  G is Tree-like iff G | _GraphSelectors in G.allSpanningTrees()
proof
  A1: G.allSpanningTrees() = G.allSpanningSG() /\ G.allTrees() by Th169;
  hereby
    assume G is Tree-like;
    then A2: G | _GraphSelectors in G.allTrees() by Th141;
    G | _GraphSelectors in G.allSpanningSG() by Th62;
    hence G | _GraphSelectors in G.allSpanningTrees()
      by A1, A2, XBOOLE_0:def 4;
  end;
  assume G | _GraphSelectors in G.allSpanningTrees();
  then G | _GraphSelectors in G.allTrees() by A1;
  hence G is Tree-like by Th141;
end;
