theorem Lem0:
  for A,B being disjoint_valued non-empty MSAlgebra over S
  for f being ManySortedFunction of A,B
  for a being Element of A holds
  the_sort_of (f.a) = the_sort_of a
  proof
    let A,B be disjoint_valued non-empty MSAlgebra over S;
    let f be ManySortedFunction of A,B;
    let a be Element of A;
    a in (the Sorts of A).the_sort_of a by SORT;
    then f.a = f.(the_sort_of a).a in (the Sorts of B).the_sort_of a
    by ABBR,FUNCT_2:5;
    hence the_sort_of (f.a) = the_sort_of a by SORT;
  end;
