reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;

theorem Th4:
  x =+=> y iff ex z st x =*=> z & z ==> y
  proof
    thus x =+=> y implies ex z st x =*=> z & z ==> y
    proof given z such that
A1:   x ==> z & z =*=> y;
      defpred P[Element of X] means ex u st x =*=> u & u ==> $1;
A2:   for y,z st y ==> z & P[y] holds P[z]
      proof
        let y,z; assume
A3:     y ==> z;
        given u such that
A4:     x =*=> u & u ==> y;
        take y;
        u =*=> y by A4,Th2;
        hence thesis by A3,A4,Th3;
      end;
A5:   for y,z st y =*=> z & P[y] holds P[z] from Star(A2);
      thus thesis by A1,A5;
    end;
    given z such that
A6: x =*=> z & z ==> y;
    defpred P[Element of X] means ex u st $1 ==> u & u =*=> y;
A2: for y,z st y ==> z & P[z] holds P[y]
    proof
      let x,z; assume
A3:   x ==> z;
      given u such that
A4:   z ==> u & u =*=> y;
      take z;
      z =*=> u by A4,Th2;
      hence thesis by A3,A4,Th3;
    end;
A5: for y,z st y =*=> z & P[z] holds P[y] from StarBack(A2);
    thus ex z st x ==> z & z =*=> y by A5,A6;
  end;
