reserve r,s,t,u for Real;

theorem Th12:
  for X being right_zeroed non empty addLoopStr, V1,V2 being
  Subset of X st 0.X in V2 holds V1 c= V1+V2
proof
  let X be right_zeroed non empty addLoopStr, V1,V2 be Subset of X such that
A1: 0.X in V2;
  let x be object;
  assume
A2: x in V1;
  then reconsider x as Point of X;
  x+0.X = x by RLVECT_1:def 4;
  then x in {u + v where u,v is Point of X: u in V1 & v in V2} by A1,A2;
  hence thesis by RUSUB_4:def 9;
end;
