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;
reserve T for non empty array of O;
reserve p,q,r,s for Element of dom T;

theorem Th73:
  for R being array of O
  st [x,y] in inversions R & inversions R is finite
  holds card inversions Swap(R,x,y) in card inversions R
  proof
    let R be array of O;
    set s = Swap(R,x,y);
    set h = (R,x,y)incl, ws = inversions s;
    assume
A1: [x,y] in inversions R & inversions R is finite; then
    reconsider w = inversions R as finite set;
    h.:ws c< inversions R by A1,Th72; then
    h.:ws c= w; then
    reconsider hws = h.:ws as finite set;
    card (hws) < card w by A1,Th72,TREES_1:6; then
A2: card (hws) in Segm card w by NAT_1:44;
A3: x in dom R & y in dom R & x in y by A1,Th46;
A4: R is non empty by A1;
    ws,h.:ws are_equipotent
    proof
      take hw = h|(ws qua set);
      thus hw is one-to-one by A3,A4,Th71;
      dom s = dom R by FUNCT_7:99; then
      ws c= [:dom R, dom R:] by Th47; then
      ws c= dom h by A3,A4,Th61;
      hence dom hw = ws by RELAT_1:62;
      thus thesis by RELAT_1:115;
    end;
    hence card inversions Swap(R,x,y) in card inversions R by A2,CARD_1:5;
  end;
