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
  for A be OSSubset of OU0 st OSConstants(OU0) (\/) A is non-empty holds
  OSMSubSort(A) is non-empty
proof
  let A be OSSubset of OU0;
  assume
A1: OSConstants(OU0) (\/) A is non-empty;
  now
    let i be object;
    assume i in the carrier of S1;
    then reconsider s = i as SortSymbol of S1;
    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;
    then
A4: (OSConstants(OU0) (\/) A).s c= meet(OSSubSort(A,s)) by SETFAM_1:5;
    ex x be object st x in (OSConstants(OU0) (\/) A).s by A1,XBOOLE_0:def 1;
    hence (OSMSubSort(A)).i is non empty by A4,Def11;
  end;
  hence thesis;
end;
