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 non empty doubleLoopStr,
  I being add-closed non empty Subset of R holds
  I is Preserv of the addF of R
  proof
    let R be non empty doubleLoopStr, I be add-closed non empty Subset of R;
    I is (the addF of R)-binopclosed
    proof
    let x be set;
    assume x in [:I,I:];
    then consider y, z being object such that
A1: y in I & z in I and
A2: x = [y,z] by ZFMISC_1:def 2;
    reconsider y, z as Element of I by A1;
    (the addF of R).x = y+z by A2;
    hence (the addF of R).x in I by IDEAL_1:def 1;
    end;
    hence thesis;
  end;
