reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;

theorem Th40:
  for U0 be non-empty OSAlgebra of S1, U1,U2 be strict
  OSSubAlgebra of U0 holds U1 /\ (U1"\/"_os U2) = U1
proof
  let U0 be non-empty OSAlgebra of S1, U1,U2 be strict OSSubAlgebra of U0;
  reconsider u1= the Sorts of U1,u2 =the Sorts of U2 as MSSubset of U0 by
MSUALG_2:def 9;
  reconsider u112=the Sorts of(U1 /\ (U1"\/"_os U2)) as MSSubset of U0 by
MSUALG_2:def 9;
A1: the Charact of (U1/\(U1"\/"_os U2))=Opers(U0,u112) by MSUALG_2:def 16;
  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 is OrderSortedSet of S1 & u2 is OrderSortedSet of S1 by OSALG_1:17;
  then A is OrderSortedSet of S1 by Th2;
  then reconsider A as OSSubset of U0 by Def2;
A2: the Sorts of (U1 /\(U1"\/"_os U2))= (the Sorts of U1)(/\)(the Sorts of(U1
  "\/"_os U2)) by MSUALG_2:def 16;
  U1"\/"_os U2 = GenOSAlg(A) by Def13;
  then A is OSSubset of U1"\/"_os U2 by Def12;
  then
A3: A c= the Sorts of (U1 "\/"_os U2) by PBOOLE:def 18;
  the Sorts of U1 c= A by PBOOLE:14;
  then the Sorts of U1 c= the Sorts of (U1"\/"_os U2) by A3,PBOOLE:13;
  then
A4: the Sorts of U1 c=the Sorts of (U1 /\(U1"\/"_os U2)) by A2,PBOOLE:17;
  the Sorts of (U1 /\(U1"\/"_os U2)) c= the Sorts of U1 by A2,PBOOLE:15;
  then the Sorts of (U1 /\(U1"\/"_os U2)) = the Sorts of U1 by A4,PBOOLE:146;
  hence thesis by A1,MSUALG_2:def 9;
end;
