
theorem Th35:
  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
    for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
    holds G1.reachableFrom(v1) = G2.reachableFrom(v2)
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();
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A2: v1 = v2;
  for w being object holds
    w in G1.reachableFrom(v1) implies w in G2.reachableFrom(v2)
  proof
    let w be object;
    assume w in G1.reachableFrom(v1);
    then consider W1 being Walk of G1 such that
      A3: W1 is_Walk_from v1,w by GLIB_002:def 5;
    W1.first() = v1 & W1.last() = w by A3, GLIB_001:def 23;
    then consider W2 being Walk of G2 such that
      A4: W2 is_Walk_from v1,w by A1;
    thus thesis by A2, A4, GLIB_002:def 5;
  end;
  then A5: G1.reachableFrom(v1) c= G2.reachableFrom(v2) by TARSKI:def 3;
  G2.reachableFrom(v2) c= G1.reachableFrom(v1) by A2, GLIB_002:14;
  hence thesis by A5, XBOOLE_0:def 10;
end;
