reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

theorem
  for R being Ring, P being Ideal of R, M being Submodule of P
  for a being BinOp of P, z be Element of P
  for m being Function of [:P,the carrier of R:], P
  st a = (the addF of R)|[:P,P:] &
  m = (the multF of R)|[:P,the carrier of R:] & z = the ZeroF of R
  holds the RightModStr of M = RightModStr(#P,a,z,m#)
  proof
    let R be Ring;
    let P be Ideal of R;
    let M be Submodule of P;
A1: the carrier of M = P by Def15;
    set V = RightModule R;
    0.M = 0.V &
    the addF of M = (the addF of V) |[:P,P:] &
    the rmult of M = (the rmult of V) | [:P, the carrier of R:]
    by A1,RMOD_2:def 2;
    hence thesis by Def15;
  end;
