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 Th23:
  for S be non void non empty ManySortedSign, U0 be non-empty
MSAlgebra over S, U1 be MSSubAlgebra of U0 holds GenMSAlg(Constants(U0)) /\ U1
  = GenMSAlg(Constants(U0))
proof
  let S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra over S
  , U1 be MSSubAlgebra of U0;
  set C = Constants(U0), G = GenMSAlg(C);
  C is MSSubset of U1 by Th10;
  then G is strict MSSubAlgebra of U1 by Def17;
  then the Sorts of G is MSSubset of U1 by Def9;
  then
A1: the Sorts of G c= the Sorts of U1 by PBOOLE:def 18;
  the Sorts of (G /\ U1) = (the Sorts of G) (/\) (the Sorts of U1) by Def16;
  then the Sorts of (G /\ U1) = the Sorts of G by A1,PBOOLE:23;
  hence thesis by Th9;
end;
