reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem Th11:
  X overlaps Y implies ex x st x in X & x in Y
proof
  deffunc F(object) = X.$1 /\ Y.$1;
  assume
A1: X overlaps Y;
A2: for i being object st i in I holds F(i) <> {} by XBOOLE_0:def 7,A1;
  consider x being ManySortedSet of I such that
A3: for i being object st i in I holds x.i in F(i)
   from KuratowskiFunction(A2);
  take x;
  now
    let i be object;
    assume i in I;
    then x.i in X.i /\ Y.i by A3;
    hence x.i in X.i by XBOOLE_0:def 4;
  end;
  hence x in X;
  now
    let i be object;
    assume i in I;
    then x.i in X.i /\ Y.i by A3;
    hence x.i in Y.i by XBOOLE_0:def 4;
  end;
  hence thesis;
end;
