theorem Th29:
  for h being ManySortedFunction of T,C
  st h is_homomorphism T,C & s = h||the generators of G
  for a being SortSymbol of S
  for t being Element of T,a
  holds t value_at(C,s) = h.a.t
  proof
    let h be ManySortedFunction of T,C;
    assume A1: h is_homomorphism T,C;
    assume A2: s = h||the generators of G;
    let a be SortSymbol of S;
    let t be Element of T,a;
A3: s is ManySortedFunction of the generators of G, the Sorts of C
    by AOFA_A00:48;
    the generators of G is_transformable_to the Sorts of C by MSAFREE4:21;
    then
A4: doms s = the generators of G by A3,MSSUBFAM:17;
    thus t value_at(C,s) = h.a.t by A4,A1,A2,AOFA_A00:def 21;
  end;
