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

theorem Th113:
  for F being PGraphMapping of G1, G2
  st F is weak_SG-embedding & rng F_V = the_Vertices_of G2
  holds rng(SG2SGFunc(F) | G1.allSpanningForests()) c= G2.allSpanningForests()
proof
  let F be PGraphMapping of G1, G2;
  set f = SG2SGFunc(F);
  assume F is weak_SG-embedding & rng F_V = the_Vertices_of G2;
  then A1: rng(f | G1.allForests()) c= G2.allForests() &
    rng(f | G1.allSpanningSG()) c= G2.allSpanningSG()
    by Th67, Th90;
  f | G1.allSpanningForests()
     = f | (G1.allSpanningSG() /\ G1.allForests()) by Th103
    .= (f | G1.allSpanningSG()) /\ (f | G1.allForests()) by RELAT_1:79;
  then A2: rng(f | G1.allSpanningForests()) c=
    rng(f|G1.allSpanningSG())/\rng(f|G1.allForests()) by RELAT_1:13;
  rng(f|G1.allSpanningSG())/\rng(f|G1.allForests()) c=
    G2.allSpanningSG() /\ G2.allForests() by A1, XBOOLE_1:27;
  then rng(f | G1.allSpanningForests()) c=
    G2.allSpanningSG() /\ G2.allForests() by A2, XBOOLE_1:1;
  hence thesis by Th103;
end;
