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 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;
