theorem Th84:
  for t being Element of T, the_array_sort_of S holds
  for t1,t2 being Element of T, I holds
  (t,t1)<-t2 value_at(C,u) =
  (t value_at(C,u), t1 value_at(C,u))<-(t2 value_at(C,u))
  proof
    let t be Element of T, the_array_sort_of S;
    let t1,t2 be Element of T, I;
    set o = In((the connectives of S).12, the carrier' of S);
    consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
A2: t2 value_at(C,u) = f.I.t2 by A1,Th28;
A3: t value_at(C,u) = f.(the_array_sort_of S).t by A1,Th28;
A4: (t,t1)<-t2 value_at(C,u) = f.(the_array_sort_of S).((t,t1)<-t2) by A1,Th28;
A5: the_arity_of o = <*the_array_sort_of S,I,I*> &
    the_result_sort_of o = the_array_sort_of S by Th76;
    then Args(o,T) = product <*(the Sorts of T).the_array_sort_of S,
    (the Sorts of T).I, (the Sorts of T).I*> by Th24;
    then reconsider p = <*t,t1,t2*> as Element of Args(o,T) by FINSEQ_3:125;
    thus (t,t1)<-t2 value_at(C,u) = Den(o,C).(f#p) by A1,A4,A5
    .= Den(o,C).<*f.(the_array_sort_of S).t,f.I.t1,f.I.t2*> by A5,Th27
    .= (t value_at(C,u), t1 value_at(C,u))<-(t2 value_at(C,u))
    by A1,A2,A3,Th28;
  end;
