
theorem
  for G1, G2 being _Graph
  for F being non empty one-to-one PGraphMapping of G1, G2
  for H2 being acyclic Subgraph of rng F
  for H1 being inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2
  holds H1 is acyclic
proof
  let G1, G2 be _Graph;
  let F be non empty one-to-one PGraphMapping of G1, G2;
  let H2 be acyclic Subgraph of rng F;
  let H1 be inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2;
  assume H1 is non acyclic;
  then consider W1 being Walk of H1 such that
    A1: W1 is Cycle-like by GLIB_002:def 2;
  reconsider W = W1 as F-defined Walk of G1 by Th106;
  W is Cycle-like by A1, GLIB_006:24;
  then A2: F.:W is Cycle-like by GLIB_010:138;
  reconsider W2 = F.:W as Walk of H2 by Th107;
  W2 is Cycle-like by A2, GLIB_006:24;
  hence contradiction by GLIB_002:def 2;
end;
