
theorem Th22:
  for G being _Graph, v1, v2 being Vertex of G st not v1 in G.reachableFrom(v2)
  holds G.reachableFrom(v1) misses G.reachableFrom(v2)
proof
  let G be _Graph, v1, v2 be Vertex of G;
  assume A1: not v1 in G.reachableFrom(v2);
  assume not G.reachableFrom(v1) misses G.reachableFrom(v2);
  then G.reachableFrom(v1) /\ G.reachableFrom(v2) <> {} by XBOOLE_0:def 7;
  then consider w being object such that
    A2: w in G.reachableFrom(v1) /\ G.reachableFrom(v2) by XBOOLE_0:def 1;
  A3: w in G.reachableFrom(v1) & w in G.reachableFrom(v2)
    by A2, XBOOLE_0:def 4;
  then consider W1 being Walk of G such that
    A4: W1 is_Walk_from v1,w by GLIB_002:def 5;
  consider W2 being Walk of G such that
    A5: W2 is_Walk_from v2,w by A3, GLIB_002:def 5;
  W1.reverse() is_Walk_from w,v1 by A4, GLIB_001:23;
  then W2.append(W1.reverse()) is_Walk_from v2,v1 by A5, GLIB_001:31;
  hence contradiction by A1, GLIB_002:def 5;
end;
