reserve x,y for set;

theorem Th15:
  for I,J being set for A being ManySortedSet of I, B being
  Function for C being ManySortedSet of J st C = Intersect(A, B) holds C cc= A
proof
  let I,J be set, A be ManySortedSet of I, B be Function;
  let C be ManySortedSet of J such that
A1: C = Intersect(A, B);
A2: dom A = I by PARTFUN1:def 2;
  dom C = J by PARTFUN1:def 2;
  then
A3: J = I /\ dom B by A1,A2,Def2;
  hence J c= I by XBOOLE_1:17;
  let i be set;
  assume i in J;
  then C.i = A.i /\ B.i by A1,A2,A3,Def2;
  hence thesis by XBOOLE_1:17;
end;
