theorem Th37:
  \1(T,I) value_at(C,s) = 1
  proof
A1: s is ManySortedFunction of the generators of G, the Sorts of C
    by AOFA_A00:48;
    then consider f being ManySortedFunction of T,C such that
A2: f is_homomorphism T,C & s = f||(the generators of G) by AOFA_A00:def 19;
    the generators of G is_transformable_to the Sorts of C
    by MSAFREE4:21;
    then
    doms s = the generators of G by A1,MSSUBFAM:17;
    then consider f being ManySortedFunction of T,C,
    Q being GeneratorSet of T such that
A3: f is_homomorphism T,C & Q = doms s & s = f||Q &
    \1(T,I) value_at(C,s) = f.I.\1(T,I) by A2,AOFA_A00:def 21;
    set o = In((the connectives of S).5, the carrier' of S);
A4: the_arity_of o = {} & the_result_sort_of o = I by Th15;
    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 A4,MSUALG_3:6;
    then
A5: p = f#p;
    f.I.\1(T,I)
    = \1(C,I) by A5,A3,A4 .= 1 by AOFA_A00:55;
    hence thesis by A3;
  end;
