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

theorem Th173:
  G is connected iff G.allSpanningTrees() <> {}
proof
  hereby
    assume G is connected;
    then reconsider G0 = G as connected _Graph;
    the plain spanning Tree-like Subgraph of G0 in G.allSpanningTrees()
      by Th168;
    hence G.allSpanningTrees() <> {};
  end;
  assume A1: G.allSpanningTrees() <> {};
  set x = the Element of G.allSpanningTrees();
  reconsider H = x as plain spanning acyclic Subgraph of G by A1, Th168;
  H is connected by A1, Th168;
  hence thesis by GLIB_002:23;
end;
