reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem Th15:
  for A be MSSubset of U0 holds Constants(U0) (\/) A c= MSSubSort(A)
proof
  let A be MSSubset of U0;
  let i be object;
  assume i in the carrier of S;
  then reconsider s = i as SortSymbol of S;
A1: for Z be set st Z in SubSort(A,s) holds (Constants(U0) (\/) A).s c= Z
  proof
    let Z be set;
    assume Z in SubSort(A,s);
    then consider B be MSSubset of U0 such that
A2: B in SubSort(A) and
A3: Z = B.s by Def13;
    Constants(U0) c= B & A c= B by A2,Th13;
    then Constants(U0) (\/) A c= B by PBOOLE:16;
    hence thesis by A3;
  end;
  (MSSubSort(A)).s = meet (SubSort(A,s)) by Def14;
  hence thesis by A1,SETFAM_1:5;
end;
