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;
reserve U0 for non-empty MSAlgebra over S;

theorem
  for S being non void non empty ManySortedSign, U0 being MSAlgebra over
  S, A being MSSubset of U0 holds SubSort(A) c= SubSort(U0)
proof
  let S be non void non empty ManySortedSign, U0 be MSAlgebra over S, A be
  MSSubset of U0;
  let x be object such that
A1: x in SubSort(A);
A2: for B be MSSubset of U0 st B = x holds B is opers_closed by A1,Def10;
  x in Funcs(the carrier of S, bool (Union (the Sorts of U0))) & x is
  MSSubset of U0 by A1,Def10;
  hence thesis by A2,Def11;
end;
