reserve r,s,t,u for Real;

theorem Th6:
  for X being add-associative non empty addLoopStr,x,y being
  Point of X, M being Subset of X holds x+y+M = x+(y+M)
proof
  let X be add-associative non empty addLoopStr,x,y be Point of X, M be
  Subset of X;
A1: x+(y+M) = {x+u where u is Point of X: u in y+M} by RUSUB_4:def 8;
A2: y+M = {y+u where u is Point of X: u in M} by RUSUB_4:def 8;
A3: x+y+M = {x+y+u where u is Point of X: u in M} by RUSUB_4:def 8;
  thus x+y+M c= x+(y+M)
  proof
    let z be object;
    assume z in x+y+M;
    then consider u being Point of X such that
A4: x+y+u = z & u in M by A3;
    x+(y+u) = z & y+u in y+M by A2,A4,RLVECT_1:def 3;
    hence thesis by A1;
  end;
  let z be object;
  assume z in x+(y+M);
  then consider u being Point of X such that
A5: x+u = z and
A6: u in y+M by A1;
  consider v being Point of X such that
A7: y+v = u and
A8: v in M by A2,A6;
  x+y+v = z by A5,A7,RLVECT_1:def 3;
  hence thesis by A3,A8;
end;
