
theorem Th33:
  for G1 being _Graph, E being Subset of the_Edges_of G1
  for G2 being inducedSubgraph of G1, the_Vertices_of G1, E
  st G2 is connected holds G1 is connected
proof
  let G1 be _Graph, E be Subset of the_Edges_of G1;
  let G2 be inducedSubgraph of G1, the_Vertices_of G1, E;
  assume A1: G2 is connected;
  now
    let v1,w1 be Vertex of G1;
    E c= the_Edges_of G1;
    then E c= G1.edgesBetween(the_Vertices_of G1) &
      the_Vertices_of G1 c= the_Vertices_of G1 by GLIB_000:34;
    then reconsider v2 = v1, w2 = w1 as Vertex of G2 by GLIB_000:def 37;
    consider W2 being Walk of G2 such that
      A2: W2 is_Walk_from v2,w2 by A1, GLIB_002:def 1;
    reconsider W1 = W2 as Walk of G1 by GLIB_001:167;
    take W1;
    thus W1 is_Walk_from v1,w1 by A2, GLIB_001:19;
  end;
  hence thesis by GLIB_002:def 1;
end;
