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 Ch7:
  X is SN implies the reduction of X is strongly-normalizing
  proof set R = the reduction of X;
    set A = the carrier of X;
A0: field R c= A \/ A by RELSET_1:8;
    assume
A1: for f being Function of NAT, A
    ex i being Nat st not f.i ==> f.(i+1);
    let f be ManySortedSet of NAT;
    per cases;
    suppose f is A-valued; then
      rng f c= A & dom f = NAT by RELAT_1:def 19,PARTFUN1:def 2; then
      reconsider g = f as Function of NAT, A by FUNCT_2:2;
      consider i being Nat such that
A2:   not g.i ==> g.(i+1) by A1;
      take i;
      thus not [f.i,f.(i+1)] in R by A2;
    end;
    suppose
      f is not A-valued; then
      consider a being object such that
A3:   a in rng f & not a in A by TARSKI:def 3,RELAT_1:def 19;
      consider i being object such that
A4:   i in dom f & a = f.i by A3,FUNCT_1:def 3;
      reconsider i as Element of NAT by A4;
      take i;
      assume [f.i,f.(i+1)] in R; then
      a in field R by A4,RELAT_1:15;
      hence thesis by A0,A3;
    end;
  end;
