  The well-known Prager-Synge identity is valid in $H^1(\Omega)$ and serves as
a foundation for developing equilibrated a posteriori error estimators for
continuous elements. In this paper, we introduce a new inequality, that may be
regarded as a generalization of the Prager-Synge identity, to be valid for
piecewise $H^1(\Omega)$ functions for diffusion problems. The inequality is
proved to be identity in two dimensions.
  For nonconforming finite element approximation of arbitrary odd order, we
propose a fully explicit approach that recovers an equilibrated flux in $H(div;
\Omega)$ through a local element-wise scheme and that recovers a gradient in
$H(curl;\Omega)$ through a simple averaging technique over edges. The resulting
error estimator is then proved to be globally reliable and locally efficient.
Moreover, the reliability and efficiency constants are independent of the jump
of the diffusion coefficient regardless of its distribution.
