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

theorem
  x =*=> y & y =+=> z implies x =+=> z
  proof
    assume
A1: x =*=> y;
    assume
A2: y =+=> z;
    consider u such that
A3: y =*=> u & u ==> z by A2,Th4;
    thus x =+=> z by A3,A1,Th3,Th4;
  end;
