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
  ex X st X is WN & X is not SN
  proof
    defpred R[set,set] means $1 = 0;
    consider X being strict non empty ARS such that
A1: the carrier of X = {0,1} and
A2: for x,y being Element of X holds x ==> y iff R[x,y] from ARSex;
    reconsider z = 0, o = 1 as Element of X by A1,TARSKI:def 2;
A3: z <> o;
    take X;
    thus X is WN
    proof
      let x be Element of X;
      x = 0 or x = 1 by A1,TARSKI:def 2; then
      x is normform or x is normalizable by A2,A3,LmA;
      hence thesis;
    end;
    set A = {z};
A4: z in A by TARSKI:def 1;
    now
      let x be Element of X;
      assume x in A; then
A5:   x = z by TARSKI:def 1;
      take y = z;
      thus y in A & x ==> y by A2,A5,TARSKI:def 1;
    end;
    hence X is not SN by A4,ThSN;
  end;
