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;

theorem Ch8:
  X is non empty & the reduction of X is strongly-normalizing implies X is SN
  proof set R = the reduction of X;
    set A = the carrier of X;
    assume
A1: X is non empty;
    assume
A5: for f being ManySortedSet of NAT
    ex i being Nat st not [f.i,f.(i+1)] in R;
    let f be Function of NAT, A;
    consider i being Nat such that
A6: not [f.i,f.(i+1)] in R by A1,A5;
    take i;
    thus not [f.i,f.(i+1)] in R by A6;
  end;
