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 Th22:
  for S be non void non empty ManySortedSign, U0 be MSAlgebra over
S, U1 be MSSubAlgebra of U0, B be MSSubset of U0 st B = the Sorts of U1
  holds GenMSAlg(B) = the MSAlgebra of U1
proof
  let S be non void non empty ManySortedSign, U0 be MSAlgebra over S, U1 be
  MSSubAlgebra of U0, B be MSSubset of U0;
  assume
A1: B = the Sorts of U1;
  set W = GenMSAlg(B);
  B is MSSubset of W by Def17;
  then the Sorts of U1 c= the Sorts of W by A1,PBOOLE:def 18;
  then
A2: U1 is MSSubAlgebra of W by Th8;
  B is MSSubset of U1 by A1,PBOOLE:def 18;
  then W is MSSubAlgebra of U1 by Def17;
  hence thesis by A2,Th7;
end;
