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 Th33:
  a,b // c,d iff PSym(p,a),PSym(p,b) // PSym(p,c),PSym(p,d)
proof
A1: now
    assume
A2: PSym(p,a),PSym(p,b) // PSym(p,c),PSym(p,d);
    d,c // PSym(p,c),PSym(p,d) by Th32;
    then d,c // PSym(p,a),PSym(p,b) by A2,Def1;
    then
A3: c,d // PSym(p,b),PSym(p,a) by Th7;
    a,b // PSym(p,b),PSym(p,a) by Th32;
    hence a,b // c,d by A3,Def1;
  end;
  now
A4: PSym(p,b),PSym(p,a) // a,b by Th3,Th32;
    assume
A5: a,b // c,d;
    PSym(p,d),PSym(p,c) // c,d by Th3,Th32;
    then PSym(p,d),PSym(p,c) // a,b by A5,Def1;
    then PSym(p,b),PSym(p,a) // PSym(p,d),PSym(p,c) by A4,Def1;
    hence PSym(p,a),PSym(p,b) // PSym(p,c),PSym(p,d) by Th7;
  end;
  hence thesis by A1;
end;
