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

theorem
  for G1, G2 being connected _Graph
  for G3 being removeLoops of G1, G4 being removeLoops of G2
  holds G3 == G4 iff G1.allSpanningTrees() = G2.allSpanningTrees()
proof
  let G1, G2 be connected _Graph;
  let G3 be removeLoops of G1, G4 be removeLoops of G2;
  hereby
    assume A1: G3 == G4;
    thus G1.allSpanningTrees() = G3.allSpanningTrees() by Th176
      .= G4.allSpanningTrees() by A1, Th177
      .= G2.allSpanningTrees() by Th176;
  end;
  assume A2: G1.allSpanningTrees() = G2.allSpanningTrees();
  G3.allSpanningTrees() = G1.allSpanningTrees() by Th176
    .= G4.allSpanningTrees() by A2, Th176;
  hence thesis by Th177;
end;
