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

theorem Th159:
  for H1, H2 being plain Tree-like Subgraph of G
  holds [H1,H2] in SubtreeRel(G) iff H1 is Subgraph of H2
proof
  let H1, H2 be plain Tree-like Subgraph of G;
  A1: H1 in G.allSG() & H2 in G.allSG() by Th1;
  hereby
    assume [H1,H2] in SubtreeRel(G);
    then [H1,H2] in SubgraphRel(G) by MMLQUER2:4;
    hence H1 is Subgraph of H2 by A1, Def6;
  end;
  A2: H1 in G.allTrees() & H2 in G.allTrees() by Th138;
  assume H1 is Subgraph of H2;
  then [H1,H2] in SubgraphRel(G) by A1, Def6;
  hence thesis by A2, MMLQUER2:4;
end;
