theorem Th51:
  [x,y] in inversions R implies [x,y] nin inversions Swap(R,x,y)
  proof assume
A1: [x,y] in inversions R; then
A2: x in dom R & y in dom R & R/.x > R/.y by Th46;
A3: not R/.x < R/.y by A1,Th46;
    Swap(R,x,y)/.x = R/.y & Swap(R,x,y)/.y = R/.x by A2,Th30,Th32;
    hence [x,y] nin inversions Swap(R,x,y) by A3,Th46;
  end;
