reserve a for set,
  i for Nat;

theorem Th13:
  for U1,U2 being Universal_Algebra st MSAlg U1 is MSSubAlgebra of
  MSAlg U2 holds the carrier of U1 is Subset of U2
proof
  let U1,U2 be Universal_Algebra;
  set MU1 = MSAlg U1, MU2 = MSAlg U2;
  assume
A1: MU1 is MSSubAlgebra of MU2;
  then reconsider MU1 as MSAlgebra over MSSign U2;
  reconsider C = the Sorts of MU1 as MSSubset of MU2 by A1,MSUALG_2:def 9;
  set gg1 = (*-->0)*(signature U2), gg2 = dom signature(U2)-->z;
  reconsider gg1 as Function of dom signature(U2), {0}* by MSUALG_1:2;
  reconsider C1 = C as ManySortedSet of the carrier of MSSign U2;
A2: 0 in {0} by TARSKI:def 1;
  MU2 = MSAlgebra(#MSSorts U2,MSCharact U2#) by MSUALG_1:def 11;
  then
A3: C1 c= MSSorts U2 by PBOOLE:def 18;
  MSSign U2 = ManySortedSign (#{0},dom signature(U2),gg1,gg2#) & MSSorts
  U2 = 0.-->the carrier of U2 by MSUALG_1:10,def 9;
  then
  MU1 = MSAlgebra(#MSSorts U1,MSCharact U1#) & C1.0 c= ({0}-->the carrier
  of U2).0 by A3,MSUALG_1:def 11;
  then (MSSorts U1).0 c= the carrier of U2 by A2,FUNCOP_1:7;
  then (0.-->the carrier of U1).0 c= the carrier of U2 by MSUALG_1:def 9;
  hence thesis by A2,FUNCOP_1:7;
end;
