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 Th26:
  for A1,A2 being MSAlgebra over S
  st the MSAlgebra of A1 = the MSAlgebra of A2
  for B1 being MSSubset of A1, B2 being MSSubset of A2
  st B1 = B2 & B1 is opers_closed holds Opers(A2,B2) = Opers(A1,B1)
  proof
    let A1,A2 be MSAlgebra over S;
    assume A1: the MSAlgebra of A1 = the MSAlgebra of A2;
    let B1 be MSSubset of A1;
    let B2 be MSSubset of A2;
    assume A2: B1 = B2 & B1 is opers_closed;
    now
      let x be object; assume x in the carrier' of S; then
      reconsider o = x as OperSymbol of S;
      thus Opers(A2,B2).x = o/.B2 by MSUALG_2:def 8
      .= o/.B1 by A1,A2,Th25
      .= Opers(A1,B1).x by MSUALG_2:def 8;
    end;
    hence Opers(A2,B2) = Opers(A1,B1);
  end;
