
theorem Th70:
  for G1 being _Graph, G2 being removeParallelEdges of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds v2.allNeighbors() = v1.allNeighbors()
proof
  let G1 be _Graph, G2 be removeParallelEdges of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  consider E being RepEdgeSelection of G1 such that
    A2: G2 is inducedSubgraph of G1, the_Vertices_of G1, E by GLIB_009:def 7;
  the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
  then E c= G1.edgesBetween(the_Vertices_of G1) &
    the_Vertices_of G1 c= the_Vertices_of G1;
  then A3: the_Edges_of G2 = E by A2, GLIB_000:def 37;
  A4: v2.allNeighbors() c= v1.allNeighbors() by A1, GLIB_000:82;
  now
    let x be object;
    assume x in v1.allNeighbors();
    then consider e0 being object such that
      A5: e0 Joins v1,x,G1 by GLIB_000:71;
    consider e being object such that
      A6: e Joins v1,x,G1 & e in E and
      for e9 being object st e9 Joins v1,x,G1 & e9 in E holds e9 = e
      by A5, GLIB_009:def 5;
    A7: x is set & e is set by TARSKI:1;
    then e Joins v1,x,G2 by A3, A6, GLIB_000:73;
    hence x in v2.allNeighbors() by A1, A7, GLIB_000:71;
  end;
  then v1.allNeighbors() c= v2.allNeighbors() by TARSKI:def 3;
  hence v2.allNeighbors() = v1.allNeighbors() by A4, XBOOLE_0:def 10;
end;
