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

theorem Th149:
  for F being PGraphMapping of G1, G2 st F is weak_SG-embedding
  holds rng(SG2SGFunc(F) | G1.allTrees()) c= G2.allTrees()
proof
  let F be PGraphMapping of G1, G2;
  set f = SG2SGFunc(F);
  f | G1.allTrees() = f | (G1.allConnectedSG() /\ G1.allForests()) by Th139
    .= (f | G1.allConnectedSG()) /\ (f | G1.allForests()) by RELAT_1:79;
  then A1: rng(f | G1.allTrees()) c=
    rng(f|G1.allConnectedSG()) /\ rng(f|G1.allForests()) by RELAT_1:13;
  assume F is weak_SG-embedding;
  then rng(f | G1.allConnectedSG()) c= G2.allConnectedSG() &
    rng(f | G1.allForests()) c= G2.allForests() by Th90, Th132;
  then rng(f | G1.allConnectedSG()) /\ rng(f | G1.allForests())
    c= G2.allConnectedSG() /\ G2.allForests() by XBOOLE_1:27;
  then rng(f|G1.allConnectedSG()) /\ rng(f|G1.allForests()) c= G2.allTrees()
    by Th139;
  hence thesis by A1, XBOOLE_1:1;
end;
