theorem
  TranslationRel S reduces s,the_sort_of C9
  proof
    defpred P[Element of Free(S,Z)] means
    TranslationRel S reduces s, the_sort_of $1;
    the_sort_of (z-term) = s by SORT;
    then
A1: P[z-term] by REWRITE1:12;
A2: now
      let o,k;
      assume
A3:   k is z-context_including;
      assume
A4:   P[z-context_in k];
      let C being context of z such that
AA:   C = o-term k;
      thus P[C]
      proof
        set t = o-term k;
        now
          take o;
          thus the_result_sort_of o = the_sort_of t by Th8;
          reconsider i = z-context_pos_in k as Element of NAT
          by ORDINAL1:def 12;
          take i;
A7:       dom k = dom the_arity_of o by MSUALG_3:6;
          hence i in dom the_arity_of o by A3,Th71;
          k.i in (the Sorts of Free(S,Z)).((the_arity_of o)/.i) &
          k.i = z-context_in k by A3,A7,Th71,MSUALG_6:2;
          hence (the_arity_of o)/.i = the_sort_of (z-context_in k)
          by SORT;
        end;
        then [the_sort_of (z-context_in k), the_sort_of (o-term k)] in
        TranslationRel S by MSUALG_6:def 3;
        then TranslationRel S reduces the_sort_of (z-context_in k),
        the_sort_of(o-term k) by REWRITE1:15;
        hence thesis by A4,AA,REWRITE1:16;
      end;
    end;
    thus P[C9] from ContextInd(A1,A2);
  end;
