reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;

theorem Th24:
  for A be OSSubset of OU0 holds OSConstants(OU0) (\/) A c= OSMSubSort(A)
proof
  let A be OSSubset of OU0;
  let i be object;
  assume i in the carrier of S1;
  then reconsider s = i as SortSymbol of S1;
A1: for Z be set st Z in OSSubSort(A,s) holds (OSConstants(OU0) (\/) A).s c= Z
  proof
    let Z be set;
    assume Z in OSSubSort(A,s);
    then consider B be OSSubset of OU0 such that
A2: B in OSSubSort(A) and
A3: Z = B.s by Def10;
    OSConstants(OU0) c= B & A c= B by A2,Th19;
    then OSConstants(OU0) (\/) A c= B by PBOOLE:16;
    hence thesis by A3;
  end;
  (OSMSubSort(A)).s = meet (OSSubSort(A,s)) by Def11;
  hence thesis by A1,SETFAM_1:5;
end;
