theorem Th62:
  for a being SortSymbol of S
  for x being pure Element of (the generators of G).a
  for u being ManySortedFunction of FreeGen T, the Sorts of C holds
  @x value_at(C,u) = u.a.x
  proof
    let a be SortSymbol of S;
    let x be pure Element of (the generators of G).a;
    let u be ManySortedFunction of FreeGen T, the Sorts of C;
    consider h being ManySortedFunction of T,C such that
A1: h is_homomorphism T,C & u = h||FreeGen T by MSAFREE4:46;
    FreeGen T is_transformable_to the Sorts of C by MSAFREE4:21;
    then
A2: doms u = FreeGen T by 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 u & u = f||Q & @x value_at(C,u) = f.a.@x
    by A1,AOFA_A00:def 21;
    @x value_at(C,u) = h.a.x by A1,A2,A3,EXTENS_1:19
    .= ((h.a)|((FreeGen T).a)).x by Def4,FUNCT_1:49;
    hence @x value_at(C,u) = u.a.x by A1,MSAFREE:def 1;
  end;
