reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;

theorem Th20:
  for F being MSAlgebra-Family of I, S, B being MSSubAlgebra of
product F for o being OperSymbol of S, x being object st x in Args(o,B)
   holds Den(o,B).x is Function & Den(o,product F).x is Function
proof
  let F be MSAlgebra-Family of I, S, B be MSSubAlgebra of product F, o be
  OperSymbol of S, x be object;
  set r = the_result_sort_of o;
  assume
A1: x in Args(o,B);
  then x in Args(o,product F) by Th18;
  then Den(o,product F).x in product Carrier(F,r) by PRALG_3:19;
  then Den(o,B).x in product Carrier(F,r) by A1,Th19;
  then reconsider p = Den(o,B).x as Element of (SORTS F).r by PRALG_2:def 10;
A2: p is Function;
  hence Den(o,B).x is Function;
  thus thesis by A1,A2,Th19;
end;
