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

theorem
  (for x,y,z st x ==> y & y ==> z holds x ==> z)
  implies
  for x,y st x =+=> y holds x ==> y
  proof
    assume
A1: for x,y,z st x ==> y & y ==> z holds x ==> z;
    let x,y;
    assume
A2: x =+=> y;
    consider z such that
A3: x ==> z and
A4: z =*=> y by A2;
    defpred P[Element of X] means x ==> $1;
A5: P[z] by A3;
A6: for u,v st u ==> v & P[u] holds P[v] by A1;
    thus P[y] from Star1(A4,A5,A6);
  end;
