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

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