reserve r,s,t,u for Real;

theorem Th11:
  for X being non empty addLoopStr, V1,V2,W1,W2 being Subset of X
  st V1 c= W1 & V2 c= W2 holds V1+V2 c= W1+W2
proof
  let X be non empty addLoopStr, V1,V2,W1,W2 be Subset of X such that
A1: V1 c= W1 & V2 c= W2;
  let x be object;
  assume x in V1+V2;
  then x in {u + v where u,v is Point of X: u in V1 & v in V2} by RUSUB_4:def 9
;
  then ex u,v being Point of X st u+v = x & u in V1 & v in V2;
  then x in {u9 + v9 where u9,v9 is Point of X: u9 in W1 & v9 in W2} by A1;
  hence thesis by RUSUB_4:def 9;
end;
