
theorem Th139:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is strong_SG-embedding
  holds G2 is acyclic implies G1 is acyclic
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is strong_SG-embedding;
  then reconsider F as non empty one-to-one PGraphMapping of G1, G2;
  assume A2: G2 is acyclic;
  not ex W1 being Walk of G1 st W1 is Cycle-like
  proof
    given W1 being Walk of G1 such that
      A3: W1 is Cycle-like;
    reconsider W1 as F-defined Walk of G1 by A1, Th121;
    F.:W1 is Cycle-like by A3, Th138;
    hence contradiction by A2, GLIB_002:def 2;
  end;
  hence G1 is acyclic by GLIB_002:def 2;
end;
