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
  PSym(p,PSym(q,a)) = PSym(q,PSym(p,a)) iff p = q or MDist p,q or MDist
  q,PSym(p,q)
proof
A1: now
    assume PSym(p,PSym(q,a))=PSym(q,PSym(p,a));
    then PSym(p,PSym(q,PSym(p,a)))=PSym(q,a) by Th29;
    then PSym(PSym(p,q),a)=PSym(q,a) by Th37;
    then q=PSym(p,q) or MDist q,PSym(p,q) by Th36;
    then
A2: Mid q,p,q or MDist q,PSym(p,q) by Def4;
    hence p = q or MDist q,p or MDist q,PSym(p,q);
    thus p = q or MDist p,q or MDist q,PSym(p,q) by A2,Th18;
  end;
  now
    assume p = q or MDist p,q or MDist q,PSym(p,q);
    then Mid q,p,q or MDist q,PSym(p,q) by Th18;
    then PSym(PSym(p,q),a)=PSym(q,a) by Def4,Th36;
    then PSym(p,PSym(q,PSym(p,a)))=PSym(q,a) by Th37;
    hence PSym(p,PSym(q,a))=PSym(q,PSym(p,a)) by Th29;
  end;
  hence thesis by A1;
end;
