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

theorem Th179:
  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.allSpanningTrees()) c= G2.allSpanningTrees()
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.allConnectedSG()) c= G2.allConnectedSG() &
    rng(f | G1.allSpanningForests()) c= G2.allSpanningForests()
    by Th132, Th113;
  f | G1.allSpanningTrees()
     = f | (G1.allSpanningForests() /\ G1.allConnectedSG()) by Th170
    .= (f|G1.allSpanningForests()) /\ (f|G1.allConnectedSG()) by RELAT_1:79;
  then A2: rng(f | G1.allSpanningTrees()) c=
    rng(f|G1.allSpanningForests())/\rng(f|G1.allConnectedSG()) by RELAT_1:13;
  rng(f|G1.allSpanningForests())/\rng(f|G1.allConnectedSG()) c=
    G2.allSpanningForests() /\ G2.allConnectedSG() by A1, XBOOLE_1:27;
  then rng(f | G1.allSpanningTrees()) c=
    G2.allSpanningForests() /\ G2.allConnectedSG() by A2, XBOOLE_1:1;
  hence thesis by Th170;
end;
