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

theorem
  for G3 being removeLoops of G1, G4 being removeLoops of G2
  holds G3 == G4 iff G1.allTrees() = G2.allTrees()
proof
  let G3 be removeLoops of G1, G4 be removeLoops of G2;
  hereby
    assume A1: G3 == G4;
    thus G1.allTrees() = G3.allTrees() by Th146
      .= G4.allTrees() by A1, Th147
      .= G2.allTrees() by Th146;
  end;
  assume A2: G1.allTrees() = G2.allTrees();
  G3.allTrees() = G1.allTrees() by Th146
    .= G4.allTrees() by A2, Th146;
  hence thesis by Th147;
end;
