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

theorem Th168:
  G2 in G1.allSpanningTrees() iff
    G2 is plain spanning acyclic Subgraph of G1 & G2 is connected
proof
  hereby
    assume G2 in G1.allSpanningTrees();
    then consider H being Element of [#]G1.allSG() such that
      A1: G2 = H & H is spanning Tree-like;
    thus G2 is plain spanning acyclic Subgraph of G1 by A1;
    thus G2 is connected by A1;
  end;
  assume A2: G2 is plain spanning acyclic Subgraph of G1 & G2 is connected;
  then G2 in [#]G1.allSG() by Th1;
  hence thesis by A2;
end;
