
theorem
  for G1 being _Graph, G2 being DSimpleGraph of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds
    v2.inNeighbors() = v1.inNeighbors() \ {v1} &
    v2.outNeighbors() = v1.outNeighbors() \ {v1} &
    v2.allNeighbors() = v1.allNeighbors() \ {v1}
proof
  let G1 be _Graph, G2 be DSimpleGraph of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  consider G9 being removeDParallelEdges of G1 such that
    A2: G2 is removeLoops of G9 by GLIB_009:120;
  reconsider v3 = v2 as Vertex of G9 by A2, GLIB_000:53;
  thus v2.inNeighbors() = v3.inNeighbors() \ {v3} by A2, Th69
    .= v1.inNeighbors() \ {v1} by A1, Th71;
  thus v2.outNeighbors() = v3.outNeighbors() \ {v3} by A2, Th69
    .= v1.outNeighbors() \ {v1} by A1, Th71;
  thus v2.allNeighbors() = v3.allNeighbors() \ {v3} by A2, Th69
    .= v1.allNeighbors() \ {v1} by A1, Th71;
end;
