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

theorem Th169:
  G.allSpanningTrees() = G.allSpanningSG() /\ G.allTrees()
proof
  now
    let x be object;
    hereby
      assume A1: x in G.allSpanningTrees();
      then reconsider H = x as plain spanning acyclic Subgraph of G by Th168;
      H is connected by A1, Th168;
      hence x in G.allSpanningSG() & x in G.allTrees() by Th60, Th138;
    end;
    assume A2: x in G.allSpanningSG() & x in G.allTrees();
    then reconsider H = x as plain spanning acyclic Subgraph of G by Th60;
    H is connected by A2;
    hence x in G.allSpanningTrees() by Th168;
  end;
  hence thesis by XBOOLE_0:def 4;
end;
