theorem
  for t1,t2 being Element of T, the bool-sort of S holds
  t1\andt2 value_at(C,u) = (t1 value_at(C,u))\and(t2 value_at(C,u))
  proof
    let t1,t2 be Element of T, the bool-sort 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: t1 value_at(C,u) = f.(the bool-sort of S).t1 by A1,Th28;
A3: t1\andt2 value_at(C,u) = f.(the bool-sort of S).(t1\andt2) by A1,Th28;
    set o = In((the connectives of S).3, the carrier' of S);
A4: the_arity_of o = <*the bool-sort of S,the bool-sort of S*> &
    the_result_sort_of o = the bool-sort of S by Th13;
    then Args(o,T) = product <*(the Sorts of T).the bool-sort of S,
    (the Sorts of T).the bool-sort of S*> by Th23;
    then reconsider p = <*t1,t2*> as Element of Args(o,T) by FINSEQ_3:124;
    thus (t1\andt2) value_at(C,u) = Den(o,C).(f#p)
    by A1,A3,A4
    .= Den(o,C).<*f.(the bool-sort of S).t1,f.(the bool-sort of S).t2*>
    by A4,Th26
    .= (t1 value_at(C,u))\and(t2 value_at(C,u)) by A2,A1,Th28;
  end;
