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 Th48:
  R is ascending iff inversions R = {}
  proof
    thus R is ascending implies inversions R = {}
    proof assume
A1:   for a,b st a in dom R & b in dom R & a in b holds R/.a <= R/.b;
      set x = the Element of inversions R;
      assume
A2:   inversions R <> {}; then x in inversions R; then
      consider a,b being Element of dom R such that
A3:   x = [a,b] & a in b & R/.a > R/.b;
      R <> {} by A2; then
      R/.a <= R/.b by A1,A3;
      hence thesis by A3,Th45;
    end;
    assume
A4: inversions R = {};
    let a,b; assume
    a in dom R & b in dom R & a in b; then
    R/.a > R/.b implies [a,b] in inversions R;
    hence thesis by A4,Th45;
  end;
