  We present an elementary and explicit proof of the separability criterion for
continuous variable two-party Gaussian systems. Our proof is based on an
elementary formulation of uncertainty relations and an explicit determination
of squeezing parameters for which the P-representation condition saturates the
$Sp(2,R)\otimes Sp(2,R)$ invariant separability condition. We thus give the
explicit formulas of squeezing parameters, which establish the equivalence of
the separability condition with the P-representation condition, in terms of the
parameters of the standard form of the correlation matrix. Our proof is
compared to the past proofs, and it is pointed out that the original proof of
the P-representation by Duan, Giedke, Cirac and Zoller(DGCZ) is incomplete. A
way to complete their proof is then shown. It is noted that both of the
corrected proof of DGCZ and the proof of R. Simon are closely related to our
explicit construction despite their quite different appearances.
