reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;

theorem
  x >><< z & z <=01= y implies x >><< y
  proof
    given u such that
A3: x =*=> u & u <=*= z;
    assume
A2: z <=01= y;
    take u;
    thus x =*=> u by A3;
    thus y =*=> u by A2,A3,Lem8;
  end;
