theorem
  \0(T,I) value_at(C,u) = 0
  proof
    consider f being ManySortedFunction of T,C such that
A1: f is_homomorphism T,C & u = f||FreeGen T by MSAFREE4:46;
    FreeGen T is_transformable_to the Sorts of C
    by MSAFREE4:21;
    then
    doms u = FreeGen T by MSSUBFAM:17;
    then consider f being ManySortedFunction of T,C,
    Q being GeneratorSet of T such that
A2: f is_homomorphism T,C & Q = doms u & u = f||Q &
    \0(T,I) value_at(C,u) = f.I.\0(T,I) by A1,AOFA_A00:def 21;
    set o = In((the connectives of S).4, the carrier' of S);
A3: the_arity_of o = {} & the_result_sort_of o = I by Th14;
    then
    Args(o,T) = {{}} by Th21;
    then reconsider p = {} as Element of Args(o,T) by TARSKI:def 1;
    dom(f#p) = {} & dom p = {} by A3,MSUALG_3:6;
    then
A4: p = f#p;
    f.I.\0(T,I) = \0(C,I) by A4,A2,A3 .= 0 by AOFA_A00:55;
    hence thesis by A2;
  end;
