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 Th13:
  for OU1 being OSSubAlgebra of OU0 holds OSConstants(OU0) is OSSubset of OU1
proof
  let OU1 be OSSubAlgebra of OU0;
  OSConstants(OU0) c= the Sorts of OU1
  proof
    let i be object;
    assume i in the carrier of S1;
    then reconsider s = i as SortSymbol of S1;
    Constants(OU0) is MSSubset of OU1 by MSUALG_2:10;
    then
A1: Constants(OU0) c= (the Sorts of OU1) by PBOOLE:def 18;
A2: for s2,s3 st s2 <= s3 holds (Constants(OU0)).s2 c= (the Sorts of OU1). s3
    proof
      let s2,s3;
      assume s2 <= s3;
      then
A3:   (the Sorts of OU1).s2 c= (the Sorts of OU1).s3 by OSALG_1:def 17;
      (Constants(OU0)).s2 c= (the Sorts of OU1).s2 by A1;
      hence thesis by A3;
    end;
A4: for X being set st X in {Constants(OU0,s1): s1 <= s} holds X c= (the
    Sorts of OU1).s
    proof
      let X be set;
      assume X in {Constants(OU0,s1): s1 <= s};
      then consider s4 such that
A5:   X = Constants(OU0,s4) & s4 <= s;
      Constants(OU0,s4) = (Constants(OU0)).s4 by MSUALG_2:def 4;
      hence thesis by A2,A5;
    end;
    (OSConstants(OU0)).i = OSConstants(OU0,s) by Def5
      .= union {Constants(OU0,s1): s1 <= s};
    hence thesis by A4,ZFMISC_1:76;
  end;
  then
A6: OSConstants(OU0) is ManySortedSubset of the Sorts of OU1 by PBOOLE:def 18;
  OSConstants(OU0) is OrderSortedSet of S1 by Def2;
  hence thesis by A6,Def2;
end;
