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 non CONF WN implies
  ex x,y,z st y is_normform_of x & z is_normform_of x & y <> z
  proof
    given a,b such that
A1: a <<>> b  & not a >><< b;
    consider x such that
A0: a <=*= x & x =*=> b by A1;
    assume
A2: c is normalizable; then
    a is normalizable; then
    consider y such that
A3: y is_normform_of a;
    b is normalizable by A2; then
    consider z such that
A4: z is_normform_of b;
    take x,y,z;
    thus y is_normform_of x & z is_normform_of x by A0,A3,A4,LemN7;
    thus thesis by A1,A3,A4;
  end;
