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 ThWN1:
  X is WN iff for x ex y st y is_normform_of x
  proof
    thus X is WN implies for x ex y st y is_normform_of x
    proof
      assume
A1:   for x holds x is normalizable;
      let x;
A2:   x is normalizable by A1;
      thus ex y st y is_normform_of x by A2;
    end;
    assume
A3: for x ex y st y is_normform_of x;
    let x; thus ex y st y is_normform_of x by A3;
  end;
