
theorem Th57:
  for G2 being _Graph, v being Vertex of G2, e,w being object
  for G1 being addAdjVertex of G2,v,e,w, v1 being Vertex of G1
  st v1 = v & not e in the_Edges_of G2 & not w in the_Vertices_of G2
  holds G1.reachableFrom(v1) = G2.reachableFrom(v) \/ {w}
proof
  let G2 be _Graph, v be Vertex of G2, e,w be object;
  let G1 be addAdjVertex of G2,v,e,w, v1 be Vertex of G1;
  assume A1: v1 = v & not e in the_Edges_of G2 & not w in the_Vertices_of G2;
  G2 is Subgraph of G1 by GLIB_006:57;
  then A2: G2.reachableFrom(v) c= G1.reachableFrom(v1) by A1, GLIB_002:14;
  A3: e Joins v1,w,G1 by A1, GLIB_006:131;
  v1 in G1.reachableFrom(v1) by GLIB_002:9;
  then {w} c= G1.reachableFrom(v1) by A3, GLIB_002:10, ZFMISC_1:31;
  then A4: G2.reachableFrom(v)\/{w} c= G1.reachableFrom(v1) by A2, XBOOLE_1:8;
  now
    let x be object;
    assume x in G1.reachableFrom(v1);
    then consider W1 being Walk of G1 such that
      A5: W1 is_Walk_from v1,x by GLIB_002:def 5;
    set P1 = the Path of W1;
    x is set by TARSKI:1;
    then A6: P1 is_Walk_from v1,x by A5, GLIB_001:160;
    then A7: P1.first() = v1 & P1.last() = x by GLIB_001:def 23;
    per cases;
    suppose A8: P1 is non trivial & x <> w;
      the_Vertices_of G1 = the_Vertices_of G2 \/ {w} by A1, GLIB_006:def 12;
      then x in the_Vertices_of G2 by A7, A8, ZFMISC_1:136;
      then not e in P1.edges() by A1, A7, GLIB_006:147;
      then reconsider P2=P1 as Walk of G2 by A1, A8, GLIB_006:145;
      P2 is_Walk_from v,x by A1, A6, GLIB_001:19;
      then x in G2.reachableFrom(v) by GLIB_002:def 5;
      hence x in G2.reachableFrom(v)\/{w} by XBOOLE_0:def 3;
    end;
    suppose P1 is trivial;
      then x = v1 by A7, GLIB_001:127;
      then x in G2.reachableFrom(v) by A1, GLIB_002:9;
      hence x in G2.reachableFrom(v)\/{w} by XBOOLE_0:def 3;
    end;
    suppose x = w;
      then x in {w} by TARSKI:def 1;
      hence x in G2.reachableFrom(v)\/{w} by XBOOLE_0:def 3;
    end;
  end;
  then G1.reachableFrom(v1) c= G2.reachableFrom(v) \/ {w} by TARSKI:def 3;
  hence thesis by A4, XBOOLE_0:def 10;
end;
