theorem
  reflection(b,reflection(a,p)) = reflection(a,reflection(b,p)) iff a = b
  proof
    reflection(b,reflection(a,p)) = reflection(a,reflection(b,p)) implies a = b
    proof
      assume
A1:   reflection(b,reflection(a,p)) = reflection(a,reflection(b,p));
      set p9 = reflection(a,p);
A2:   Middle p,a,p9 by DEFR;
      Middle reflection(b,p),a,reflection(b,p9) by A1,DEFR;
      then Middle reflection(b,reflection(b,p)),reflection(b,a),
         reflection(b,reflection(b,p9)) by Satz7p15,Satz7p16;
      then Middle p,reflection(b,a),reflection(b,reflection(b,p9)) by Satz7p7;
      then Middle p,reflection(b,a),p9 by Satz7p7;
      hence thesis by A2,Satz7p17,Satz7p10;
    end;
    hence thesis;
  end;
