
theorem
  for R being non empty addLoopStr, I,J being Subset of R, K being
  add-closed Subset of R st I c= K & J c= K holds I + J c= K
proof
  let R be non empty addLoopStr, I,J be Subset of R, K being add-closed (
  Subset of R);
  assume
A1: I c= K & J c= K;
    let u be object;
    assume u in I + J;
    then ex i,j being Element of R st u = i + j & i in I & j in J;
    hence u in K by A1,Def1;
end;
