
theorem Th83:
  for F being non empty Graph-yielding Function, x being Element of dom F
  holds (the_Vertices_of canGFDistinction F).x
    = [: {[the_Vertices_of F,x]}, (the_Vertices_of F).x :]
proof
  let F be non empty Graph-yielding Function, x be Element of dom F;
  reconsider x9 = x as Element of dom canGFDistinction F by Def25;
  thus (the_Vertices_of canGFDistinction F).x
     = the_Vertices_of ((canGFDistinction F).x9) by Def8
    .= the_Vertices_of replaceVerticesEdges(
        renameElementsDistinctlyFunc(the_Vertices_of F,x),
        renameElementsDistinctlyFunc(the_Edges_of F,x)) by Def25
    .= rng renameElementsDistinctlyFunc(the_Vertices_of F,x) by Th1
    .= [: {[the_Vertices_of F,x]}, (the_Vertices_of F).x :] by Th80;
end;
