reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem Th27:
  for S be non void non empty ManySortedSign, U0 be non-empty
  MSAlgebra over S, U1,U2 be MSSubAlgebra of U0 holds U1 /\ (U1"\/"U2) =
  the MSAlgebra of U1
proof
  let S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra over S
  , U1,U2 be MSSubAlgebra of U0;
  reconsider u1= the Sorts of U1,u2 =the Sorts of U2 as MSSubset of U0 by Def9;
A1: the Sorts of (U1 /\(U1"\/"U2))
   =(the Sorts of U1) (/\) (the Sorts of(U1"\/"U2)) by Def16;
  u1 c= the Sorts of U0 & u2 c= the Sorts of U0 by PBOOLE:def 18;
  then u1 (\/) u2 c= the Sorts of U0 by PBOOLE:16;
  then reconsider A= u1 (\/) u2 as MSSubset of U0 by PBOOLE:def 18;
  U1"\/"U2 = GenMSAlg(A) by Def18;
  then A is MSSubset of U1"\/"U2 by Def17;
  then
A2: A c= the Sorts of (U1 "\/" U2) by PBOOLE:def 18;
  the Sorts of U1 c= A by PBOOLE:14;
  then the Sorts of U1 c= the Sorts of (U1"\/"U2) by A2,PBOOLE:13;
  then
A3: the Sorts of U1 c=the Sorts of (U1 /\(U1"\/"U2)) by A1,PBOOLE:17;
  reconsider u112=the Sorts of(U1 /\ (U1"\/"U2)) as MSSubset of U0 by Def9;
A4: the Charact of (U1/\(U1"\/" U2))=Opers(U0,u112) by Def16;
  the Sorts of (U1 /\(U1"\/"U2)) c= the Sorts of U1 by A1,PBOOLE:15;
  then the Sorts of (U1 /\(U1"\/"U2)) = the Sorts of U1 by A3,PBOOLE:146;
  hence thesis by A4,Def9;
end;
