reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem
  pi(X /\ Y,x) c= pi(X,x) /\ pi(Y,x)
proof
  let y be object;
  assume y in pi(X /\ Y,x);
  then consider f such that
A1: f in X /\ Y and
A2: y = f.x by Def6;
A3: f in X by A1,XBOOLE_0:def 4;
A4: f in Y by A1,XBOOLE_0:def 4;
A5: y in pi(X,x) by A2,A3,Def6;
  y in pi(Y,x) by A2,A4,Def6;
  hence thesis by A5,XBOOLE_0:def 4;
end;
