reserve X,Y,x for set;
reserve A for non empty preBoolean set;

theorem
  for A, B being set holds Fin (A /\ B) = Fin A /\ Fin B
proof
  let A, B be set;
  Fin (A /\ B) c= Fin A & Fin (A /\ B) c= Fin B by Th10,XBOOLE_1:17;
  hence Fin (A /\ B) c= Fin A /\ Fin B by XBOOLE_1:19;
    let X be object;
    reconsider XX=X as set by TARSKI:1;
    assume
A1: X in Fin A /\ Fin B;
    then X in Fin B by XBOOLE_0:def 4;
    then
A2: XX c= B by Def5;
A3: X in Fin A by A1,XBOOLE_0:def 4;
    then XX c= A by Def5;
    then XX c= A /\ B by A2,XBOOLE_1:19;
    hence X in Fin (A /\ B) by A3,Def5;
end;
