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 LemN12:
  X is WN UN* & x is normform & x <=*=> y implies y =*=> x
  proof
    assume
A1: X is WN UN*;
    assume
A2: x is normform;
    assume
A3: x <=*=> y;
    defpred P[Element of X] means $1 =*=> x;
A4: for y,z st y <==> z & P[y] holds P[z]
    proof
      let y,z;
      assume
B1:   y <==> z;
      assume
B2:   P[y];
      per cases by B1;
      suppose
C1:     y ==> z;
B3:     z is normalizable by A1;
        consider u such that
B4:     u is_normform_of z by B3;
B5:     u is_normform_of y by C1,B4,LemN6;
B6:     x is_normform_of y by A2,B2;
        thus P[z] by B4,B6,B5,A1;
      end;
      suppose
C2:     y <== z;
        thus P[z] by B2,C2,Lem5;
      end;
    end;
A5: for y,z st y <=*=> z & P[y] holds P[z] from Star2(A4);
    thus y =*=> x by A3,A5;
  end;
