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
  X is CR iff the reduction of X is with_Church-Rosser_property
  proof
    set R = the reduction of X;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    thus X is CR implies R is with_Church-Rosser_property
    proof assume
A1:   x <=*=> y implies x >><< y;
      let a,b be object; assume
A2:   a,b are_convertible_wrt R;
      per cases by A2,REWRITE1:32;
      suppose
        a in field R & b in field R; then
        reconsider x = a, y = b as Element of X by F0;
        x <=*=> y by A2;
        hence a,b are_convergent_wrt R by A1,Ch12;
      end;
      suppose
        a = b;
        hence a,b are_convergent_wrt R by REWRITE1:38;
      end;
    end;
    assume
A5: for a,b being object st a,b are_convertible_wrt R
    holds a,b are_convergent_wrt R;
    let x,y; assume x <=*=> y;
    hence thesis by A5,Ch12;
  end;
