theorem Th14:
  for A,B being disjoint_valued non-empty MSAlgebra over S
  for C being non-empty MSAlgebra over S
  for f being ManySortedFunction of A,B
  for g being ManySortedFunction of B,C
  for a being Element of A
  holds (g**f).a = g.(f.a)
  proof
    let A,B be disjoint_valued non-empty MSAlgebra over S;
    let C be non-empty MSAlgebra over S;
    let f be ManySortedFunction of A,B;
    let g be ManySortedFunction of B,C;
    let a be Element of A;
A1: a in (the Sorts of A).the_sort_of a &
    f.(the_sort_of a) is Function of (the Sorts of A).(the_sort_of a),
    (the Sorts of B).(the_sort_of a) by SORT;
    thus (g**f).a = (g**f).(the_sort_of a).a by ABBR
    .= ((g.the_sort_of a)*(f.(the_sort_of a))).a by MSUALG_3:2
    .= (g.the_sort_of a).(f.(the_sort_of a).a) by A1,FUNCT_2:15
    .= (g.the_sort_of (f.a)).(f.(the_sort_of a).a) by Lem0
    .= (g.the_sort_of (f.a)).(f.a) by ABBR
    .= g.(f.a) by ABBR;
  end;
