reserve a for set,
  i for Nat;

theorem Th9:
  for U1,U2 being Universal_Algebra st U1 is SubAlgebra of U2 holds
  the Sorts of MSAlg U1 is MSSubset of MSAlg U2
proof
  let U1,U2 be Universal_Algebra;
  set gg1 = (*-->0)*(signature U2), gg2 = dom signature(U2)-->z;
  reconsider gg1 as Function of dom signature(U2), {0}* by MSUALG_1:2;
A1: MSSign U2 = ManySortedSign (#{0},dom signature(U2),gg1,gg2#) by MSUALG_1:10
;
  MSAlg U2 = MSAlgebra(#MSSorts U2,MSCharact U2#) by MSUALG_1:def 11;
  then
A2: the Sorts of MSAlg U2 = 0 .--> the carrier of U2 by MSUALG_1:def 9;
  assume
A3: U1 is SubAlgebra of U2;
  then MSSign U1 = MSSign U2 by Th7;
  then reconsider A = MSAlg U1 as non-empty MSAlgebra over MSSign U2;
  MSAlg U1 = MSAlgebra(#MSSorts U1,MSCharact U1#) by MSUALG_1:def 11;
  then
A4: the Sorts of A = 0 .--> the carrier of U1 by MSUALG_1:def 9;
A5: 0 in {0} by TARSKI:def 1;
A6: the carrier of U1 is Subset of U2 by A3,UNIALG_2:def 7;
  the Sorts of A c= the Sorts of MSAlg U2
  proof
    let i be object;
    assume i in the carrier of MSSign U2;
    then
A7: i = 0 by A1,TARSKI:def 1;
    (the Sorts of A).0 = the carrier of U1 & (the Sorts of MSAlg U2).0 =
    the carrier of U2 by A4,A2,A5,FUNCOP_1:7;
    hence thesis by A6,A7;
  end;
  hence thesis by PBOOLE:def 18;
end;
