 Rademacher span is not complemented in L1

If the closed span of the Rademacher functions were a complemented subspace of $L^1(0,1)$, compose a projection with the inverse of its l2 sequence space parametrization. The Grothendieck theorem for L1 to Hilbert operators would make this map an absolutely summing operator. Its composition with the parametrization would make the identity on the infinite-dimensional l2 sequence space absolutely summing, contradicting its coordinate-vector test.