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

theorem
  x <==> y iff [x,y] in (the reduction of X)\/(the reduction of X)~
  proof
A1: x ==> y iff [x,y] in the reduction of X;
A2: x <== y iff [y,x] in the reduction of X;
    [y,x] in the reduction of X iff [x,y] in (the reduction of X)~
    by RELAT_1:def 7;
    hence thesis by A1,A2,XBOOLE_0:def 3;
  end;
