reserve x,y for set;

theorem
  for I being set, A,B being ManySortedSet of I
    holds Intersect(A, B) = A (/\) B
proof
  let I be set, A,B be ManySortedSet of I;
A1: dom A = I & dom B = I by PARTFUN1:def 2;
  then dom Intersect(A,B) = I /\ I by Def2;
  then reconsider AB = Intersect(A,B) as ManySortedSet of I by PARTFUN1:def 2
,RELAT_1:def 18;
  I /\ I = I;
  then for i being object st i in I holds AB.i = A.i /\ B.i by A1,Def2;
  hence thesis by PBOOLE:def 5;
end;
