theorem Th29:
  for z1 being Element of z holds TrivialOSA(S,z,z1) is non-empty
  & TrivialOSA(S,z,z1) is monotone
proof
  let z1 be Element of z;
  set A = TrivialOSA(S,z,z1);
  the Sorts of A = ConstOSSet(S,z) by Def22;
  then
A1: the Sorts of A is non-empty by Th15;
  hence A is non-empty by MSUALG_1:def 3;
  reconsider A1 = A as non-empty OSAlgebra of S by A1,MSUALG_1:def 3;
  for o1,o2 st o1 <= o2 holds Den(o1,A1) c= Den(o2,A1)
  proof
    let o1,o2;
A2: Args(o1,A) = ((the Sorts of A)# * the Arity of S).o1 by MSUALG_1:def 4
      .= (the Sorts of A)#.((the Arity of S).o1) by FUNCT_2:15
      .= (the Sorts of A)#.(the_arity_of o1) by MSUALG_1:def 1;
A3: Args(o2,A) = ((the Sorts of A)# * the Arity of S).o2 by MSUALG_1:def 4
      .= (the Sorts of A)#.((the Arity of S).o2) by FUNCT_2:15
      .= (the Sorts of A)#.(the_arity_of o2) by MSUALG_1:def 1;
    assume o1 <= o2;
    then
A4: (the_arity_of o1) <= (the_arity_of o2);
    Den(o1,A) = Args(o1,A) --> z1 & Den(o2,A) = Args(o2,A) --> z1 by Def22;
    hence thesis by A4,A2,A3,Th20,FUNCT_4:4;
  end;
  hence thesis by Th27;
end;
