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 Th46:
 for x being object, y being set holds
  [x,y] in inversions R iff x in dom R & y in dom R & x in y & R/.x > R/.y
  proof let x be object, y be set;
    hereby
      assume
A1:   [x,y] in inversions R; then
      reconsider R1 = R as non empty Function;
      consider a,b being Element of dom R1 such that
A2:   [x,y] = [a,b] & a in b & not R/.a <= R/.b by A1;
      x = a & y = b by A2,XTUPLE_0:1;
      hence x in dom R & y in dom R & x in y & R/.x > R/.y by A2,Th45;
    end;
    thus thesis;
  end;
