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 Th21:
  for S be non void non empty ManySortedSign, U0 be
MSAlgebra over S, B be MSSubset of U0 st B = the Sorts of U0 holds GenMSAlg(B)
  = the MSAlgebra of U0
proof
  let S be non void non empty ManySortedSign, U0 be MSAlgebra over S, B
  be MSSubset of U0;
  set W = GenMSAlg(B);
  reconsider B1 = the Sorts of W as MSSubset of U0 by Def9;
A1: the Charact of W = Opers(U0,B1) by Def9;
  assume B = the Sorts of U0;
  then the Sorts of U0 is MSSubset of W by Def17;
  then
A2: the Sorts of U0 c= the Sorts of W by PBOOLE:def 18;
  (the Sorts of W) is MSSubset of U0 by Def9;
  then the Sorts of W c= the Sorts of U0 by PBOOLE:def 18;
  then the Sorts of U0 = the Sorts of W by A2,PBOOLE:146;
  hence thesis by A1,Th4;
end;
