reserve
  S for (4,1) integer bool-correct non empty non void BoolSignature,
  X for non-empty ManySortedSet of the carrier of S,
  T for vf-free integer all_vars_including inheriting_operations free_in_itself
  (X,S)-terms VarMSAlgebra over S,
  C for (4,1) integer bool-correct non-empty image of T,
  G for basic GeneratorSystem over S,X,T,
  A for IfWhileAlgebra of the generators of G,
  I for integer SortSymbol of S,
  x,y,z,m for pure (Element of (the generators of G).I),
  b for pure (Element of (the generators of G).the bool-sort of S),
  t,t1,t2 for Element of T,I,
  P for Algorithm of A,
  s,s1,s2 for Element of C-States(the generators of G);
reserve
  f for ExecutionFunction of A, C-States(the generators of G),
  (\falseC)-States(the generators of G, b);
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;

theorem Th28:
  for h being ManySortedFunction of T,C st h is_homomorphism T,C
  for a being SortSymbol of S
  for t being Element of T,a
  holds t value_at(C,h||FreeGen T) = h.a.t
  proof
    let h be ManySortedFunction of T,C;
    assume A1: h is_homomorphism T,C;
    set s = h||FreeGen T;
    let a be SortSymbol of S;
    let t be Element of T,a;
    FreeGen T is_transformable_to the Sorts of C by MSAFREE4:21;
    then
A2: doms s = FreeGen T by MSSUBFAM:17;
    thus t value_at(C,s) = h.a.t by A2,A1,AOFA_A00:def 21;
  end;
