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 Th37:
  for U0 be non-empty OSAlgebra of S1, U1 be OSSubAlgebra of U0, A
,B be OSSubset of U0 st B = A (\/) the Sorts of U1
  holds GenOSAlg(A) "\/"_os U1 = GenOSAlg(B)
proof
  let U0 be non-empty OSAlgebra of S1, U1 be OSSubAlgebra of U0, A,B be
  OSSubset of U0;
  assume
A1: B = A (\/) the Sorts of U1;
  reconsider u1m = the Sorts of U1, am = the Sorts of GenOSAlg(A) as MSSubset
  of U0 by MSUALG_2:def 9;
A2: the Sorts of U1 is OrderSortedSet of S1 & the Sorts of GenOSAlg(A) is
  OrderSortedSet of S1 by OSALG_1:17;
  then reconsider u1 = u1m, a = am as OSSubset of U0 by Def2;
  a c= the Sorts of U0 & u1 c= the Sorts of U0 by PBOOLE:def 18;
  then a (\/) u1 c= the Sorts of U0 by PBOOLE:16;
  then reconsider b=a (\/) u1 as MSSubset of U0 by PBOOLE:def 18;
A3: a (\/) u1 is OrderSortedSet of S1 by A2,Th2;
  then reconsider b as OSSubset of U0 by Def2;
A4: (GenOSAlg(A) "\/"_os U1) = GenOSAlg(b) by Def13;
  then a (\/) u1 is OSSubset of (GenOSAlg(A)"\/"_os U1) by Def12;
  then
A5: a (\/) u1 c=the Sorts of (GenOSAlg(A)"\/"_os U1) by PBOOLE:def 18;
  A is OSSubset of GenOSAlg(A) by Def12;
  then
A6: A c= the Sorts of GenOSAlg(A) by PBOOLE:def 18;
  then A (\/) u1 c= a (\/) u1 by PBOOLE:20;
  then B c=the Sorts of (GenOSAlg(A)"\/"_os U1) by A1,A5,PBOOLE:13;
  then
A7: B is MSSubset of (GenOSAlg(A)"\/"_os U1) by PBOOLE:def 18;
A8: A is OrderSortedSet of S1 by Def2;
A9: the Sorts of (GenOSAlg(A) /\ GenOSAlg(B)) = (the Sorts of GenOSAlg(A))
  (/\) (the Sorts of GenOSAlg(B)) by MSUALG_2:def 16;
  B is OSSubset of GenOSAlg(B) by Def12;
  then
A10: B c= the Sorts of GenOSAlg(B) by PBOOLE:def 18;
  A c= B by A1,PBOOLE:14;
  then A c= the Sorts of GenOSAlg(B) by A10,PBOOLE:13;
  then A c= (the Sorts of GenOSAlg(A)) (/\) (the Sorts of GenOSAlg(B))
     by A6,PBOOLE:17;
  then A is MSSubset of (GenOSAlg(A) /\ GenOSAlg(B)) by A9,PBOOLE:def 18;
  then A is OSSubset of (GenOSAlg(A) /\ GenOSAlg(B)) by A8,Def2;
  then GenOSAlg(A) is OSSubAlgebra of (GenOSAlg(A) /\ GenOSAlg(B)) by Def12;
  then a is MSSubset of (GenOSAlg(A) /\ GenOSAlg(B)) by MSUALG_2:def 9;
  then
A11: a c= (the Sorts of GenOSAlg(A)) (/\) (the Sorts of GenOSAlg(B)) by A9,
PBOOLE:def 18;
  (the Sorts of GenOSAlg(A)) (/\) (the Sorts of GenOSAlg(B)) c= a by PBOOLE:15;
  then a= (the Sorts of GenOSAlg(A)) (/\) (the Sorts of GenOSAlg(B)) by A11,
PBOOLE:146;
  then
A12: a c= the Sorts of GenOSAlg(B) by PBOOLE:15;
  u1 c= B by A1,PBOOLE:14;
  then u1 c= the Sorts of GenOSAlg(B) by A10,PBOOLE:13;
  then b c= the Sorts of GenOSAlg(B) by A12,PBOOLE:16;
  then b is MSSubset of GenOSAlg(B) by PBOOLE:def 18;
  then b is OSSubset of GenOSAlg(B) by A3,Def2;
  then
A13: GenOSAlg(b) is strict OSSubAlgebra of GenOSAlg(B) by Def12;
  B is OrderSortedSet of S1 by Def2;
  then B is OSSubset of (GenOSAlg(A)"\/"_os U1) by A7,Def2;
  then GenOSAlg(B) is strict OSSubAlgebra of (GenOSAlg(A)"\/"_os U1) by Def12;
  hence thesis by A4,A13,MSUALG_2:7;
end;
