theorem Th43:
  for w,p being Element of Args(o,Free(S,Z)) for t being Element of Free(S,Z)
  st w is z-context_including & C9 = o-term w &
  p = w+*(z-context_pos_in w, (z-context_in w)-sub(t)) &
  the_sort_of t = s
  holds C9-sub(t) = o-term p
  proof
    let w,p be Element of Args(o,Free(S,Z));
    let t be Element of Free(S,Z);
    assume that
A1: w is z-context_including and
A2: C9 = o-term w and
A3: p = w+*(z-context_pos_in w, (z-context_in w)-sub(t)) and
A4: the_sort_of t = s;
A5: dom p = dom the_arity_of o = dom w <> {} by A1,MSUALG_3:6;
    then reconsider v = w, q = p as non empty DTree-yielding FinSequence;
    now let i,d1; assume
A6:   i in dom v & d1 = v.i;
A7:   (z-context_in w)-sub(t) = (z-context_in w,[z,s])<-t by A4,SUB;
      per cases;
      suppose
A8:     i = z-context_pos_in w;
        then d1 = z-context_in w by A1,A6,Th71;
        hence q.i = (d1,[z,s])<-t by A3,A6,A7,A8,FUNCT_7:31;
      end;
      suppose
A8:     i <> z-context_pos_in w;
        then w/.i is z-omitting & w/.i = d1 by A1,A6,Th72,PARTFUN1:def 6;
        then (d1,[z,s])<-t = d1 by Th23;
        hence q.i = (d1,[z,s])<-t by A3,A6,A8,FUNCT_7:32;
      end;
    end;
    then ([o,the carrier of S]-tree v,[z,s])<-t
    = [o,the carrier of S]-tree q by A5,ThL8;
    hence C9-sub(t) = o-term p by A2,A4,SUB;
  end;
