
theorem
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  for V2 being non empty Subset of the_Vertices_of rng F
  for H2 being inducedSubgraph of rng F, V2
  for H1 being inducedSubgraph of G1,F_V"the_Vertices_of H2
  st G1.edgesBetween(F_V"the_Vertices_of H2) c= dom F_E holds rng(F | H1) == H2
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let V2 be non empty Subset of the_Vertices_of rng F;
  let H2 be inducedSubgraph of rng F, V2;
  let H1 be inducedSubgraph of G1,F_V"the_Vertices_of H2;
  assume A1: G1.edgesBetween(F_V"the_Vertices_of H2) c= dom F_E;
  A2: H2 is Subgraph of G2 by GLIB_000:43;
  A3: F_E"the_Edges_of H2 = G1.edgesBetween(F_V"the_Vertices_of H2)
    by A1, Th101;
  set v = the Vertex of H2;
  v in the_Vertices_of H2;
  then v in the_Vertices_of rng F;
  then v in rng F_V by GLIB_010:54;
  then consider x being object such that
    A4: x in dom F_V & F_V.x = v by FUNCT_1:def 3;
  A5: x in F_V"the_Vertices_of H2 by A4, FUNCT_1:def 7;
  then A6: the_Vertices_of H1 = F_V"the_Vertices_of H2 &
    the_Edges_of H1 = F_E"the_Edges_of H2 by A3, GLIB_000:def 37;
  the_Vertices_of H2 c= the_Vertices_of rng F;
  then A7: the_Vertices_of H2 c= rng F_V by GLIB_010:54;
  the_Edges_of H2 c= the_Edges_of rng F;
  then A8: the_Edges_of H2 c= rng F_E by GLIB_010:54;
  x in (dom F_V) /\ the_Vertices_of H1 by A6, A4, A5, XBOOLE_0:def 4;
  then x in dom(F | H1)_V by RELAT_1:61;
  then A9: F | H1 is non empty;
  then A10: the_Vertices_of rng(F | H1)
     = rng(F_V | the_Vertices_of H1) by GLIB_010:54
    .= F_V.:(F_V"the_Vertices_of H2) by A6, RELAT_1:115
    .= the_Vertices_of H2 by A7, FUNCT_1:77;
  the_Edges_of rng(F | H1)
     = rng(F_E | the_Edges_of H1) by A9, GLIB_010:54
    .= F_E.:(F_E"the_Edges_of H2) by A6, RELAT_1:115
    .= the_Edges_of H2 by A8, FUNCT_1:77;
  hence thesis by A2, A10, GLIB_000:86;
end;
