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

theorem Th69:
  for F being PGraphMapping of G1, G2 st F is isomorphism
  holds rng(SG2SGFunc(F) | G1.allSpanningSG()) = G2.allSpanningSG()
proof
  let F be PGraphMapping of G1, G2;
  assume F is isomorphism;
  then F is onto & F_V is one-to-one & dom F_V = the_Vertices_of G1
    by GLIB_010:def 11;
  then F is onto & F_V is one-to-one & F_V is total by PARTFUN1:def 2;
  hence thesis by Th68;
end;
