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 Th35:
  for OU1 being strict OSSubAlgebra of OU0, B being OSSubset of
  OU0 st B = the Sorts of OU1 holds GenOSAlg(B) = OU1
proof
  let OU1 be strict OSSubAlgebra of OU0;
  let B be OSSubset of OU0;
  set W = GenOSAlg(B);
  assume
A1: B = the Sorts of OU1;
  then
A2: B is MSSubset of OU1 by PBOOLE:def 18;
  B is OSSubset of W by Def12;
  then the Sorts of OU1 c= the Sorts of W by A1,PBOOLE:def 18;
  then
A3: OU1 is strict MSSubAlgebra of W by MSUALG_2:8;
  B is OrderSortedSet of S1 by Def2;
  then B is OSSubset of OU1 by A2,Def2;
  then W is strict OSSubAlgebra of OU1 by Def12;
  hence thesis by A3,MSUALG_2:7;
end;
