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 Th32:
  for A being OSSubset of OU0 holds the Sorts of GenMSAlg(A) c=
  the Sorts of GenOSAlg(A)
proof
  let A be OSSubset of OU0;
  set GM = GenMSAlg(A), GO = GenOSAlg(A);
  let i be object;
  assume i in the carrier of S1;
  then reconsider s = i as SortSymbol of S1;
  the Sorts of GM = MSSubSort(A) by Th31;
  then
A1: (the Sorts of GM).s = meet SubSort(A,s) by MSUALG_2:def 14;
  the Sorts of GO = OSMSubSort(A) by Th30;
  then (the Sorts of GO).s = meet OSSubSort(A,s) by Def11;
  hence thesis by A1,Th22,SETFAM_1:6;
end;
