theorem Th38:
  (-t) value_at(C,s) = -(t value_at(C,s))
  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
A3: doms s = the generators of G by A1,MSSUBFAM:17;
    then consider f1 being ManySortedFunction of T,C,
    Q1 being GeneratorSet of T such that
A4: f1 is_homomorphism T,C & Q1 = doms s & s = f1||Q1 &
    t value_at(C,s) = f1.I.t by A2,AOFA_A00:def 21;
    consider f2 being ManySortedFunction of T,C,
    Q2 being GeneratorSet of T such that
A5: f2 is_homomorphism T,C & Q2 = doms s & s = f2||Q2 &
    (-t) value_at(C,s) = f2.I.(-t) by A2,A3,AOFA_A00:def 21;
    set o = In((the connectives of S).6, the carrier' of S);
A6: the_arity_of o = <*I*> & the_result_sort_of o = I by Th16;
    then Args(o,T) = product <*(the Sorts of T).I*> by Th22;
    then reconsider p = <*t*> as Element of Args(o,T) by FINSEQ_3:123;
    thus (-t) value_at(C,s) = Den(o,C).(f2#p) by A5,A6
    .= Den(o,C).<*f2.I.t*> by A6,Th25
    .= -(t value_at(C,s)) by A4,A5,EXTENS_1:19;
  end;
