reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;
reserve I for non empty set,
  x,X,Y for ManySortedSet of I;
reserve I for set,
  x,X,Y,Z for ManySortedSet of I;
reserve X for non-empty ManySortedSet of I;

theorem
  (for x holds x in X iff x in Y & x in Z) implies X = Y (/\) Z
proof
  assume
A1: for x holds x in X iff x in Y & x in Z;
  now
    let x;
    hereby
      assume x in X;
      then x in Y & x in Z by A1;
      hence x in Y (/\) Z by Th8;
    end;
    assume x in Y (/\) Z;
    then x in Y & x in Z by Th8;
    hence x in X by A1;
  end;
  hence thesis by Th135;
end;
