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 Th71:
  p in q implies (T,p,q)incl|(inversions Swap(T,p,q) qua set) is one-to-one
  proof assume
A1: p in q;
    set f = (T,p,q)incl;
    set s = Swap(T,p,q);
    set w = inversions s;
    set fw = f|(w qua set);
A2: dom s = dom T by FUNCT_7:99;
    let x,y be object; assume x in dom fw & y in dom fw; then
A3: x in w & y in w by RELAT_1:57; then
    consider x1,y1 being Element of dom T such that
A4: x = [x1,y1] & x1 in y1 & s/.x1 > s/.y1 by A2;
    consider x2,y2 being Element of dom T such that
A5: y = [x2,y2] & x2 in y2 & s/.x2 > s/.y2 by A2,A3;
    assume fw.x = fw.y; then
    f.(x1,y1) = fw.y by A4,A3,FUNCT_1:49 .= f.(x2,y2) by A5,A3,FUNCT_1:49; then
    x1 = x2 & y1 = y2 by A1,A4,A5,Lm1;
    hence thesis by A4,A5;
  end;
