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

theorem
  (for x,y st x ==> y holds y ==> x)
  implies
  for x,y st x <=*=> y holds x =*=> y
  proof
    assume
A:  for x,y st x ==> y holds y ==> x;
    let x,y;
    assume
B:  x <=*=> y;
    defpred P[Element of X] means x =*=> $1;
C:  for u,v st u <==> v & P[u] holds P[v] by A,Lem10;
D:  for u,v st u <=*=> v & P[u] holds P[v] from Star2(C);
    thus x =*=> y by B,D;
end;
