The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} P_{i+1,j,J} + P_{i,i+1,J}P_{j,j+1,J}. \quad (1) \end{align*} is a subsystem of the system of Plucker relations: $i,j,k,l \in \mathbb{Z}_{\ge 1}$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,k,l\}=\emptyset$, $i<j<l<k$, \begin{align*} P_{i,k,J}P_{j,l,J} = P_{i,l,J} P_{j,k,J} + P_{i,j,J}P_{k,l,J}. \quad (2) \end{align*}

Do the equations in (2) follow from the equation in (1)? Thank you very much.

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} P_{i+1,j,J} + P_{i,i+1,J}P_{j,j+1,J}. \quad (1) \end{align*} is a subsystem of the system of Plucker relations: $i,j,k,l \in \mathbb{Z}_{\ge 1}$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,k,l\}=\emptyset$, $i<j<l<k$, \begin{align*} P_{i,k,J}P_{j,l,J} = P_{i,l,J} P_{j,k,J} + P_{i,j,J}P_{k,l,J}. \quad (2) \end{align*}

Do the equations in (2) follow from the equation in (1)? Thank you very much.

Edit: division by positive integers is allowed, division by P's is allowed, division by polynomials of P's is allowed.