theorem Th50:
  [x,y] in inversions R & [y,z] in inversions R implies [x,z] in inversions R
  proof assume
A1: [x,y] in inversions R & [y,z] in inversions R; then
    reconsider x,y,z as Element of dom R by Th46;
    x in y & R/.x > R/.y & y in z & R/.y > R/.z by A1,Th46; then
    x in z & R/.x > R/.z by ORDERS_2:5,ORDINAL1:10;
    hence thesis;
  end;
