reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;

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;
