
theorem
  for G1, G2 being _Graph
  for F being non empty one-to-one PGraphMapping of G1, G2
  for H2 being connected Subgraph of rng F
  for H1 being inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2
  holds H1 is connected
proof
  let G1, G2 be _Graph;
  let F be non empty one-to-one PGraphMapping of G1, G2;
  let H2 be connected Subgraph of rng F;
  let H1 be inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2;
  now
    let v1, w1 be Vertex of H1;
    H2 is Subgraph of G2 by GLIB_000:43;
    then A1: F_E"the_Edges_of H2 c= G1.edgesBetween(F_V"the_Vertices_of H2)
      by Th99;
    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
      A2: x in dom F_V & F_V.x = v by FUNCT_1:def 3;
    x in F_V"the_Vertices_of H2 by A2, FUNCT_1:def 7;
    then the_Vertices_of H1 = F_V"the_Vertices_of H2 &
      the_Edges_of H1 = F_E"the_Edges_of H2 by A1, GLIB_000:def 37;
    then A3: v1 in dom F_V & F_V.v1 in the_Vertices_of H2 &
      w1 in dom F_V & F_V.w1 in the_Vertices_of H2 by FUNCT_1:def 7;
    then consider W2 being Walk of H2 such that
      A4: W2 is_Walk_from F_V.v1, F_V.w1 by GLIB_002:def 1;
    reconsider W3 = W2 as F-valued Walk of G2 by Th108;
    reconsider W1 = F"W3 as Walk of H1 by Th109;
    take W1;
    W3 is_Walk_from F_V.v1, F_V.w1 by A4, GLIB_001:19;
    then F"W3 is_Walk_from F"_V.(F_V.v1), F"_V.(F_V.w1) by Th97;
    then W1 is_Walk_from F"_V.(F_V.v1), F"_V.(F_V.w1) by GLIB_001:19;
    then W1 is_Walk_from v1, F_V".(F_V.w1) by A3, FUNCT_1:34;
    hence W1 is_Walk_from v1,w1 by A3, FUNCT_1:34;
  end;
  hence thesis by GLIB_002:def 1;
end;
