Hi, I found the following relationship: it is given x is a vector, bm1 and bm2 are vectors of type logical end exactly as long as x then the following formula returns alwasy TRUE: all( x[ bm1 & bm2] == x[ bm2][ bm1[ bm2]] ) . I think it is no such great deal and a lot of R programmers will know this and use it in their code intutively. But the point is: Years ago I tried to prove this relationship and found no acceptable solution for that. Is anyone able to prove this in a mathematical correct way? Besides that: here is another one it is given a dataframe 'result_df' with at least the column 'result_value' best_results <- with( result_df, sort( result_value))[1:100] # lower is better best_results_bmask <- with( result_df, result_value %in% best_results) best_results_order2 <- with( result_df[ best_results_bmask, ], order( result_value)) df1 <- result_df[best_results_order, ][ best_results_bmask[best_results_order], ] df2 <- result_df[best_results_bmask, ][best_results_order2, ] all( df1 == df2) # that was the easy part, now this: df3 <- result_df[best_results_order[ best_results_bmask[ best_results_order]], ] all( df1 == df3) # it works, but how can I understand that and how it is proven? Regards Frank Schwidom