reserve r,s,t,u for Real;

theorem Th8:
  for X being non empty addLoopStr, M,N being Subset of X, x being
  Point of X st M c= N holds x+M c= x+N
proof
  let X be non empty addLoopStr, M,N be Subset of X, x be Point of X such that
A1: M c= N;
  let z be object;
  assume
A2: z in x+M;
  then reconsider z as Point of X;
  x+M = {x + u where u is Element of X: u in M} by RUSUB_4:def 8;
  then ex u being Point of X st x+u = z & u in M by A2;
  hence thesis by A1,Lm1;
end;
