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
  ex d st PSym(a,PSym(b,PSym(c,p))) = PSym(d,p)
proof
  consider e such that
A1: Mid a,e,c by Th19;
  consider d such that
A2: Mid b,e,d by Th21;
  c = PSym(e,a) by A1,Def4;
  then PSym(c,PSym(d,p)) = PSym(PSym(e,a),PSym(PSym(e,b),p)) by A2,Def4
    .= PSym(PSym(e,a),PSym(e,PSym(b,PSym(e,p)))) by Th37
    .= PSym(e,PSym(a,PSym(e,PSym(e,PSym(b,PSym(e,p)))))) by Th37
    .= PSym(e,PSym(a,PSym(b,PSym(e,p)))) by Th29
    .= PSym(e,PSym(e,PSym(b,PSym(a,p)))) by Th39
    .= PSym(b,PSym(a,p)) by Th29;
  then PSym(d,p) = PSym(c,PSym(b,PSym(a,p))) by Th29;
  hence thesis by Th39;
end;
