reserve AFV for WeakAffVect;
reserve a,b,c,d,e,f,a9,b9,c9,d9,f9,p,q,r,o,x99 for Element of AFV;

theorem Th39:
  PSym(p,PSym(q,PSym(r,a))) = PSym(r,PSym(q,PSym(p,a)))
proof
  p,a // PSym(r,a),PSym(r,p) & PSym(q,PSym(r,p)),PSym(q,PSym(r,a)) //
  PSym(r,a ),PSym(r,p) by Th3,Th32;
  then
A1: p,a // PSym(q,PSym(r,p)),PSym(q,PSym(r,a)) by Def1;
  p,a // PSym(p,a),PSym(p,p) & PSym(q,PSym(p,p)),PSym(q,PSym(p,a)) //
  PSym(p,a ),PSym(p,p) by Th3,Th32;
  then
A2: p,a // PSym(q,PSym(p,p)),PSym(q,PSym(p,a)) by Def1;
  PSym(q,p),PSym(r,p) // PSym(r,PSym(r,p)),PSym(r,PSym(q,p)) by Th32;
  then PSym(q,p),PSym(r,p) // p,PSym(r,PSym(q,p)) by Th29;
  then
A3: p,PSym(r,PSym(q,p)) // PSym(q,p),PSym(r,p) by Th3;
  PSym(q,PSym(r,p)),p // PSym(q,p),PSym(q,PSym(q,PSym(r,p))) by Th32;
  then PSym(q,PSym(r,p)),p // PSym(q,p),PSym(r,p) by Th29;
  then PSym(q,PSym(r,p)),p // p,PSym(r,PSym(q,p)) by A3,Def1;
  then Mid PSym(q,PSym(r,p)),p,PSym(r,PSym(q,p));
  then PSym(p,PSym(q,PSym(r,p))) = PSym(r,PSym(q,p)) by Def4;
  then
A4: PSym(p,PSym(q,PSym(r,p))) = PSym(r,PSym(q,PSym(p,p))) by Th28;
  PSym(r,PSym(q,PSym(p,a))),PSym(r,PSym(q,PSym(p,p))) // PSym(q,PSym(p,p)
  ),PSym(q,PSym(p,a)) by Th3,Th32;
  then
A5: PSym(r,PSym(q,PSym(p,a))),PSym(r,PSym(q,PSym(p,p))) // p,a by A2,Def1;
  PSym(p,PSym(q,PSym(r,a))),PSym(p,PSym(q,PSym(r,p))) // PSym(q,PSym(r,p)
  ),PSym(q,PSym(r,a)) by Th3,Th32;
  then PSym(p,PSym(q,PSym(r,a))),PSym(p,PSym(q,PSym(r,p))) // p,a by A1,Def1;
  then PSym(p,PSym(q,PSym(r,a))),PSym(p,PSym(q,PSym(r,p))) // PSym(r,PSym(q,
  PSym(p,a))),PSym(p,PSym(q,PSym(r,p))) by A4,A5,Def1;
  hence thesis by Th4,Th7;
end;
