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.allSpanningForests() = G2.allSpanningForests()
proof
  let G3 be removeLoops of G1, G4 be removeLoops of G2;
  hereby
    assume A1: G3 == G4;
    thus G1.allSpanningForests() = G3.allSpanningForests() by Th110
      .= G4.allSpanningForests() by A1, Th111
      .= G2.allSpanningForests() by Th110;
  end;
  assume A2: G1.allSpanningForests() = G2.allSpanningForests();
  G3.allSpanningForests() = G1.allSpanningForests() by Th110
    .= G4.allSpanningForests() by A2, Th110;
  hence thesis by Th111;
end;
