reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th34:
  for T being non-empty trivial MSAlgebra over S
  for A being non-empty MSSubAlgebra of T holds
  the MSAlgebra of A = the MSAlgebra of T
  proof
    let T be non-empty trivial MSAlgebra over S;
    let A be non-empty MSSubAlgebra of T;
A1: the Sorts of A is ManySortedSubset of the Sorts of T
    by MSUALG_2:def 9;
A2: now
      let x be object;
      assume
A3:   x in the carrier of S; then
A4:   (the Sorts of A).x c= (the Sorts of T).x & (the Sorts of A).x <> {} &
      (the Sorts of T).x <> {} by A1,PBOOLE:def 2,def 18;
      (the Sorts of A).x is non empty trivial by A4;
      then consider a being object such that
A5:   (the Sorts of A).x = {a} by ZFMISC_1:131;
      consider b being object such that
A6:   (the Sorts of T).x = {b} by A3,ZFMISC_1:131;
      thus (the Sorts of A).x = (the Sorts of T).x
      by A4,A5,A6,ZFMISC_1:3;
    end;
    the MSAlgebra of T = the MSAlgebra of T;
    then T is MSSubAlgebra of T & A is MSSubAlgebra of T by MSUALG_2:5;
    hence the MSAlgebra of A = the MSAlgebra of T by A2,PBOOLE:3,MSUALG_2:9;
  end;
