reserve x,y for set;

theorem
  for A, B being category st A, B have_the_same_composition & Intersect(
A,B) is non empty & for a being Object of A, b being Object of B st a = b holds
  idm a = idm b holds Intersect(A, B) is subcategory of A
proof
  let A,B be category such that
A1: A, B have_the_same_composition and
A2: Intersect(A,B) is non empty and
A3: for a being Object of A, b being Object of B st a = b holds idm a = idm b;
  reconsider AB = Intersect(A,B) as transitive non empty SubCatStr of A by A1
,A2,Th20,Th22;
A4: the carrier of AB = (the carrier of A) /\ (the carrier of B) by A1,Def3;
  now
    let o be Object of AB, a be Object of A;
    reconsider b = o as Object of B by A4,XBOOLE_0:def 4;
    assume
A5: o = a;
    then idm a = idm b by A3;
    hence idm a in <^o,o^> by A1,A5,Th23;
  end;
  hence thesis by ALTCAT_2:def 14;
end;
