
theorem Th71:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds
    v2.inNeighbors() = v1.inNeighbors() &
    v2.outNeighbors() = v1.outNeighbors() &
    v2.allNeighbors() = v1.allNeighbors()
proof
  let G1 be _Graph, G2 be removeDParallelEdges of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  consider E being RepDEdgeSelection of G1 such that
    A2: G2 is inducedSubgraph of G1, the_Vertices_of G1, E by GLIB_009:def 8;
  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.inNeighbors() c= v1.inNeighbors() by A1, GLIB_000:82;
  now
    let x be object;
    assume x in v1.inNeighbors();
    then consider e0 being object such that
      A5: e0 DJoins x,v1,G1 by GLIB_000:69;
    consider e being object such that
      A6: e DJoins x,v1,G1 & e in E and
      for e9 being object st e9 DJoins x,v1,G1 & e9 in E holds e9 = e
      by A5, GLIB_009:def 6;
    A7: x is set & e is set by TARSKI:1;
    then e DJoins x,v1,G2 by A3, A6, GLIB_000:73;
    hence x in v2.inNeighbors() by A1, A7, GLIB_000:69;
  end;
  then v1.inNeighbors() c= v2.inNeighbors() by TARSKI:def 3;
  hence A8: v2.inNeighbors() = v1.inNeighbors() by A4, XBOOLE_0:def 10;
  A9: v2.outNeighbors() c= v1.outNeighbors() by A1, GLIB_000:82;
  now
    let x be object;
    assume x in v1.outNeighbors();
    then consider e0 being object such that
      A10: e0 DJoins v1,x,G1 by GLIB_000:70;
    consider e being object such that
      A11: e DJoins v1,x,G1 & e in E and
      for e9 being object st e9 DJoins v1,x,G1 & e9 in E holds e9 = e
      by A10, GLIB_009:def 6;
    A12: x is set & e is set by TARSKI:1;
    then e DJoins v1,x,G2 by A3, A11, GLIB_000:73;
    hence x in v2.outNeighbors() by A1, A12, GLIB_000:70;
  end;
  then v1.outNeighbors() c= v2.outNeighbors() by TARSKI:def 3;
  hence A13: v2.outNeighbors() = v1.outNeighbors() by A9, XBOOLE_0:def 10;
  thus v2.allNeighbors() = v1.allNeighbors() by A8, A13;
end;
