reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th13:
  for I being set
  for X being ManySortedSet of I
  for S being ManySortedSubset of X
  for x holds S.x is Subset of X.x
  proof
    let I be set;
    let X be ManySortedSet of I;
    let S be ManySortedSubset of X;
    let x;
A1: S c= X by PBOOLE:def 18;
    x in dom S & dom S = I or x nin dom S by PARTFUN1:def 2; then
    S.x c= X.x or S.x = {} by A1,FUNCT_1:def 2;
    hence thesis by XBOOLE_1:2;
  end;
