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

theorem Th103:
  G.allSpanningForests() = G.allSpanningSG() /\ G.allForests()
proof
  now
    let x be object;
    hereby
      assume x in G.allSpanningForests();
      then x is plain spanning acyclic Subgraph of G by Th102;
      hence x in G.allSpanningSG() & x in G.allForests() by Th60, Th78;
    end;
    assume x in G.allSpanningSG() & x in G.allForests();
    then x is plain spanning Subgraph of G & x is plain acyclic Subgraph of G
      by Th60;
    hence x in G.allSpanningForests() by Th102;
  end;
  hence thesis by XBOOLE_0:def 4;
end;
