theorem
  ex x st x in X
proof
  deffunc F(object) = X.$1;
A1: for i being object st i in I holds F(i) <> {} by Def13;
  consider f being ManySortedSet of I such that
A2: for i being object st i in I holds f.i in F(i)
from KuratowskiFunction(A1);
  take f;
  let i be object;
  thus thesis by A2;
end;
