reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem Th24:
  for A being non-empty MSAlgebra over S for F being
  ManySortedFunction of A, Trivial_Algebra S for o being OperSymbol of S for x
  being Element of Args(o,A) holds (F.the_result_sort_of o).(Den(o,A).x) = 0 &
  Den(o,Trivial_Algebra S).(F#x) = 0
proof
  let A be non-empty MSAlgebra over S, F be ManySortedFunction of A,
  Trivial_Algebra S;
  let o be OperSymbol of S;
  let x be Element of Args(o,A);
  set I = the carrier of S, SA = the Sorts of A, T = Trivial_Algebra S, ST =
  the Sorts of T;
  set r = the_result_sort_of o;
  consider i being object such that
A1: i in I and
A2: Result(o,T) = ST.i by PBOOLE:138;
  reconsider d = Den(o,A).x as Element of SA.r by FUNCT_2:15;
  consider XX being ManySortedSet of I such that
A3: {XX} = I --> {0} by Th5;
A4: ST = {XX} by A3,MSAFREE2:def 12;
  then
A5: ST.r = {0} by A3;
  thus (F.r).(Den(o,A).x) = (F.r).d .= 0 by A5,TARSKI:def 1;
  ST.i = {0} by A3,A4,A1,FUNCOP_1:7;
  hence thesis by A2,TARSKI:def 1;
end;
