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
  for A1,A2,B being MSAlgebra over S
  st the MSAlgebra of A1 = the MSAlgebra of A2
  for B1 being MSSubAlgebra of A1 st the MSAlgebra of B = the MSAlgebra of B1
  holds B is MSSubAlgebra of A2
  proof
    let A1,A2,B be MSAlgebra over S;
    assume A1: the MSAlgebra of A1 = the MSAlgebra of A2;
    let B1 be MSSubAlgebra of A1;
    assume A2: the MSAlgebra of B = the MSAlgebra of B1;
    thus the Sorts of B is MSSubset of A2 by A1,A2,MSUALG_2:def 9;
    let C be MSSubset of A2;
    reconsider C1 = C as MSSubset of A1 by A1;
    assume
A3: C = the Sorts of B;
    hence C is opers_closed by A1,Th27,A2,MSUALG_2:def 9;
    the Charact of B1 = Opers(A1,C1) by A2,A3,MSUALG_2:def 9;
    hence the Charact of B = Opers(A2,C) by A1,A2,A3,MSUALG_2:def 9,Th26;
  end;
