
theorem Th101:
  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 H being inducedSubgraph of rng F, V2
  st G1.edgesBetween(F_V"the_Vertices_of H) c= dom F_E
  holds F_E"the_Edges_of H = G1.edgesBetween(F_V"the_Vertices_of H)
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 H be inducedSubgraph of rng F, V2;
  assume A1: G1.edgesBetween(F_V"the_Vertices_of H) c= dom F_E;
  H is Subgraph of G2 by GLIB_000:43;
  then A2: F_E"the_Edges_of H c= G1.edgesBetween(F_V"the_Vertices_of H)
    by Th99;
  now
    let e be object;
    set v = (the_Source_of G1).e, w = (the_Target_of G1).e;
    assume A3: e in G1.edgesBetween(F_V"the_Vertices_of H);
    then A4: e in dom F_E by A1;
    A5: e in the_Edges_of G1 & v in F_V"the_Vertices_of H &
      w in F_V"the_Vertices_of H by A3, GLIB_000:31;
    then A6: v in dom F_V & F_V.v in the_Vertices_of H & w in dom F_V &
      F_V.w in the_Vertices_of H by FUNCT_1:def 7;
    e Joins v,w,G1 by A5, GLIB_000:def 13;
    then A7: F_E.e Joins F_V.v,F_V.w,G2 by A4, A6, GLIB_010:4;
    F_E.e in rng F_E by A4, FUNCT_1:3;
    then F_E.e in the_Edges_of rng F by GLIB_010:54;
    then F_E.e Joins F_V.v,F_V.w,rng F by A7, GLIB_000:73;
    then F_E.e in (rng F).edgesBetween(the_Vertices_of H) by A6, GLIB_000:32;
    then F_E.e in (rng F).edgesBetween(V2) by GLIB_000:def 37;
    then F_E.e in the_Edges_of H by GLIB_000:def 37;
    hence e in F_E"the_Edges_of H by A4, FUNCT_1:def 7;
  end;
  then G1.edgesBetween(F_V"the_Vertices_of H) c= F_E"the_Edges_of H
    by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
