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 <=01=> y implies x >>01<< y
  proof
    assume
A1: x <=01=> y;
    per cases by A1,Lem31;
    suppose x = y;
      hence thesis;
    end;
    suppose
A2:   x ==> y;
      take y;
      thus x =01=> y & y <=01= y by A2;
    end;
    suppose
A3:   x <== y;
      take x;
      thus x =01=> x & x <=01= y by A3;
    end;
  end;
