reserve r,s,t,u for Real;

theorem Th10:
  for X being non empty addLoopStr, M,N,V being Subset of X st M
  c= N holds M+V c= N+V
proof
  let X be non empty addLoopStr, M,N,V be Subset of X such that
A1: M c= N;
  let x be object;
  assume x in M+V;
  then x in {u + v where u,v is Point of X: u in M & v in V} by RUSUB_4:def 9;
  then ex u,v being Point of X st u+v = x & u in M & v in V;
  then x in {u9 + v9 where u9,v9 is Point of X: u9 in N & v9 in V} by A1;
  hence thesis by RUSUB_4:def 9;
end;
