reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;

theorem Th26:
  for A, B being strict non-empty finitely-generated MSSubAlgebra
of U0 ex M being strict non-empty finitely-generated MSSubAlgebra of U0 st A is
  MSSubAlgebra of M & B is MSSubAlgebra of M
proof
  let A, B be strict non-empty finitely-generated MSSubAlgebra of U0;
  len <%A,B%> = 2 by AFINSQ_1:38;
  then dom <%A,B%> = {0,1} by CARD_1:50;
  then reconsider F = <%A,B%> as ManySortedSet of {0,1}
    by RELAT_1:def 18, PARTFUN1:def 2;
A1: F.1 = B;
A2: F.0 = A;
  F is MSAlgebra-Family of {0,1}, S
  proof
    let i be object;
    assume i in {0,1};
    hence thesis by A1,TARSKI:def 2,A2;
  end;
  then reconsider F as MSAlgebra-Family of {0,1}, S;
  for i being Element of {0,1} ex C being strict non-empty
  finitely-generated MSSubAlgebra of U0 st C = F.i
  proof
    let i be Element of {0,1};
    per cases by TARSKI:def 2;
    suppose
A3:   i = 0;
      take A;
      thus thesis by A3;
    end;
    suppose
A4:   i = 1;
      take B;
      thus thesis by A4;
    end;
  end;
  then consider
  M being strict non-empty finitely-generated MSSubAlgebra of U0 such
  that
A5: for i being Element of {0,1} holds F.i is MSSubAlgebra of M by Th25;
  take M;
   0 in {0,1} by TARSKI:def 2;
  hence A is MSSubAlgebra of M by A5,A2;
   1 in {0,1} by TARSKI:def 2;
  hence thesis by A5,A1;
end;
