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 Lem27:
  X is WN UN* & x <=*=> y implies nf x = nf y
  proof
    assume
A1: X is WN UN*;
    assume
A2: x <=*=> y;
    defpred P[Element of X] means nf x = nf $1;
A3: P[x];
A4: for z,u st z <==> u & P[z] holds P[u] by A1,Th2,Lem26;
    P[y] from Star2A(A2,A3,A4);
    hence thesis;
  end;
