reserve S for non void non empty ManySortedSign,
  U1, U2, U3 for non-empty MSAlgebra over S,
  I for set,
  A for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for U1 being MSAlgebra over S for A, B being MSSubset of U1 st A is
  ManySortedSubset of B holds GenMSAlg A is MSSubAlgebra of GenMSAlg B
proof
  let U1 be MSAlgebra over S, A, B be MSSubset of U1;
  B is MSSubset of GenMSAlg B by MSUALG_2:def 17;
  then
A1: B c= the Sorts of GenMSAlg B by PBOOLE:def 18;
  assume A is ManySortedSubset of B;
  then A c= B by PBOOLE:def 18;
  then A c= the Sorts of GenMSAlg B by A1,PBOOLE:13;
  then A is MSSubset of GenMSAlg B by PBOOLE:def 18;
  hence thesis by MSUALG_2:def 17;
end;
