theorem
  for t1,t2 being Element of T, I holds
  init.array(t1,t2) value_at(C,u) =
  init.array(t1 value_at(C,u), t2 value_at(C,u))
  proof
    let t1,t2 be Element of T, I;
    set o = In((the connectives of S).14, 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: init.array(t1,t2) value_at(C,u) = f.(the_array_sort_of S)
    .(init.array(t1,t2)) by A1,Th28;
A4: the_arity_of o = <*I,I*> &
    the_result_sort_of o = the_array_sort_of S by Th78;
    then Args(o,T) = product <*(the Sorts of T).I, (the Sorts of T).I*>
    by Th23;
    then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
    thus (init.array(t1,t2)) value_at(C,u) = Den(o,C).(f#p)
    by A1,A3,A4
    .= Den(o,C).<*f.I.t1,f.I.t2*> by A4,Th26
    .= init.array(t1 value_at(C,u),t2 value_at(C,u)) by A1,A2,Th28;
  end;
