theorem Th56:
  [x,y] in inversions R & x in z & [z,y] in inversions Swap(R,x,y) implies
  [z,y] in inversions R
  proof assume
    [x,y] in inversions R; then
A1: x in dom R & y in dom R & x in y & R/.x > R/.y by Th46;
A2: dom Swap(R,x,y) = dom R by FUNCT_7:99;
    assume
A3: x in z;
    assume
    [z,y] in inversions Swap(R,x,y); then
A4: z in dom R & z in y & Swap(R,x,y)/.z > Swap(R,x,y)/.y by A2,Th46; then
    z <> x & z <> y by A3; then
    Swap(R,x,y)/.y = R/.x & Swap(R,x,y)/.z = R/.z by A1,A4,Th32,Th34; then
    R/.z > R/.y by A1,A4,ORDERS_2:5;
    hence [z,y] in inversions R by A1,A4;
  end;
