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

theorem LemZ:
  x =*=> y implies x <=*=> y
  proof
    assume
A1: x =*=> y;
    defpred P[Element of X] means x <=*=> $1;
A2: P[x];
A3: for y,z st y ==> z & P[y] holds P[z]
    proof
      let y,z;
      assume
A4:   y ==> z;
      assume
A5:   P[y];
A6:   y <==> z by A4;
A7:   y <=*=> z by A6,Th6;
      thus P[z] by A5,A7,Th7;
    end;
    thus P[y] from Star1(A1,A2,A3);
  end;
