
theorem Th36:
  for G1 being _Graph, G2 being Subgraph of G1
  st (for W1 being Walk of G1 ex W2 being Walk of G2
    st W2 is_Walk_from W1.first(),W1.last())
  holds G1 is connected implies G2 is connected
proof
  let G1 be _Graph, G2 be Subgraph of G1;
  assume A1: for W1 being Walk of G1 ex W2 being Walk of G2
    st W2 is_Walk_from W1.first(),W1.last();
  assume A2: G1 is connected;
  now
    let v2,w2 be Vertex of G2;
    the_Vertices_of G2 c= the_Vertices_of G1;
    then reconsider v1=v2, w1=w2 as Vertex of G1 by TARSKI:def 3;
    consider W1 being Walk of G1 such that
      A3: W1 is_Walk_from v1,w1 by A2, GLIB_002:def 1;
    W1.first() = v1 & W1.last() = w1 by A3, GLIB_001:def 23;
    then consider W2 being Walk of G2 such that
      A4: W2 is_Walk_from v1,w1 by A1;
    take W2;
    thus W2 is_Walk_from v2,w2 by A4;
  end;
  hence thesis by GLIB_002:def 1;
end;
