theorem
  C9 is basic implies transl C9 is_e.translation_of Free(S,Z),s,the_sort_of C9
  proof set X = Z, C = C9, x = z;
    given o,k such that
Z0: C = o-term k & x-context_in k = z-term;
    set p = k;
    take o;
    thus the_result_sort_of o = the_sort_of C by Z0,Th8;
    reconsider i = x-context_pos_in p as Element of NAT by ORDINAL1:def 12;
    take i;
A1: p is x-context_including by Z0,Th53;
    then i in dom p by Th71;
    hence i in dom the_arity_of o by MSUALG_3:6;
    then x-term = p.i in (the Sorts of Free(S,X)).((the_arity_of o)/.i)
    by Z0,A1,Th71,MSUALG_6:2;
    then
A4: s = the_sort_of (x-context_in p) = (the_arity_of o)/.i by Z0,SORT;
    hence ((the_arity_of o)/.i) = s;
    reconsider a = p as Function;
    take a; thus a in Args(o,Free(S,X));
    dom transl C = (the Sorts of Free(S,X)).s by FUNCT_2:def 1;
    hence dom transl C = dom transl(o,i,a,Free(S,X)) by A4,MSUALG_6:def 4;
    let c be object; assume
    c in dom transl C;
    then reconsider c as Element of (the Sorts of Free(S,X)).s;
    reconsider q = p+*(i,c) as Element of Args(o,Free(S,X)) by A4,MSUALG_6:7;
A6: transl(o,i,a,Free(S,X)).c = Den(o,Free(S,X)).q by A4,MSUALG_6:def 4
    .= o-term q by MSAFREE4:13;
    C-sub c = o-term q & the_sort_of c = s by Z0,Th58,SORT;
    hence thesis by A6,TRANS;
  end;
