reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;

theorem Lem19:
  x <<>> y implies x <=*=> y
  proof
    given u such that
A2: x <=*= u & u =*=> y;
A3: x <=*=> u & u <=*=> y by A2,LemZ;
    thus x <=*=> y by A3,Th7;
  end;
