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

theorem Th177:
  for G1, G2 being loopless connected _Graph
  holds G1 == G2 iff G1.allSpanningTrees() = G2.allSpanningTrees()
proof
  let G1, G2 be loopless connected _Graph;
  hereby
    assume G1 == G2;
    then G2 is removeLoops of G1 by GLIB_009:58;
    hence G1.allSpanningTrees() = G2.allSpanningTrees() by Th176;
  end;
  assume G1.allSpanningTrees() = G2.allSpanningTrees();
  then G1 is Subgraph of G2 & G2 is Subgraph of G1 by Th175;
  hence thesis by GLIB_000:87;
end;
