reserve G for _Graph;
reserve V for non empty set, E for Relation of V;
reserve E for symmetric Relation of V;
reserve G for GraphFromSymRel of V, E;

theorem
  for v being Vertex of G holds Im(E,v) = v.allNeighbors()
proof
  let v be Vertex of G;
  set G0 = createGraph(V,E);
  consider E0 being RepEdgeSelection of G0 such that
    A1: G is inducedSubgraph of G0, the_Vertices_of G0, E0 by GLIB_009:def 7;
  A2: the_Edges_of G0 = G0.edgesBetween(the_Vertices_of G0) by GLIB_000:34;
  the_Vertices_of G0 c= the_Vertices_of G0;
  then reconsider v0 = v as Vertex of G0 by A1, A2, GLIB_000:def 37;
  thus Im(E,v) = Im(E,v) \/ Im(E,v)
    .= Im(E,v) \/ Coim(E,v) by GLIBPRE0:10
    .= v0.outNeighbors() \/ Coim(E,v) by Th72
    .= v0.outNeighbors() \/ v0.inNeighbors() by Th73
    .= v0.allNeighbors() by GLIB_000:def 48
    .= v.allNeighbors() by GLIBPRE0:64;
end;
