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

theorem Th91:
  for F being PGraphMapping of G1, G2 st F is one-to-one onto
  holds G2.allForests() c= rng(SG2SGFunc(F) | G1.allForests())
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is one-to-one onto;
  set f = SG2SGFunc(F) | G1.allForests();
  A2: dom f = G1.allForests() by FUNCT_2:def 1;
  now
    let x be object;
    assume x in G2.allForests();
    then reconsider H2 = x as plain acyclic Subgraph of G2 by Th78;
    rng F == G2 by A1, GLIB_010:56;
    then A3: H2 is Subgraph of rng F by GLIB_000:91;
    set H1 = the plain inducedSubgraph of
      G1, F_V"the_Vertices_of H2, F_E"the_Edges_of H2;
    H1 is acyclic by A1, A3, GLIBPRE1:107;
    then A4: H1 in dom f by A2, Th78;
    A5: rng(F | H1) = H2 by A1, A3, GLIB_009:44, GLIBPRE1:99;
    f.H1 = SG2SGFunc(F).H1 by A4, FUNCT_1:47
      .= H2 by A5, Def5;
    hence x in rng f by A4, FUNCT_1:3;
  end;
  hence thesis by TARSKI:def 3;
end;
