reserve r,s,t,u for Real;

theorem Th7:
  for X being add-associative non empty addLoopStr, x being Point
  of X, M,N being Subset of X holds x+M+N = x+(M+N)
proof
  let X be add-associative non empty addLoopStr, x be Point of X, M,N be
  Subset of X;
A1: x+(M+N) = {x+u where u is Point of X: u in M+N} by RUSUB_4:def 8;
A2: x+M = {x+u where u is Point of X: u in M} by RUSUB_4:def 8;
A3: M+N = {u+v where u,v is Point of X: u in M & v in N} by RUSUB_4:def 9;
A4: x+M+N = {u+v where u,v is Point of X: u in x+M & v in N} by RUSUB_4:def 9;
  thus x+M+N c= x+(M+N)
  proof
    let z be object;
    assume z in x+M+N;
    then consider u,v being Point of X such that
A5: u+v = z and
A6: u in x+M and
A7: v in N by A4;
    consider u9 being Point of X such that
A8: x+u9 = u & u9 in M by A2,A6;
    x+(u9+v) = z & u9+v in M+N by A3,A5,A7,A8,RLVECT_1:def 3;
    hence thesis by A1;
  end;
  let z be object;
  assume z in x+(M+N);
  then consider u being Point of X such that
A9: x+u = z and
A10: u in M+N by A1;
  consider w,v being Point of X such that
A11: w+v = u and
A12: w in M and
A13: v in N by A3,A10;
A14: x+w in x+M by A2,A12;
  x+w+v = z by A9,A11,RLVECT_1:def 3;
  hence thesis by A4,A13,A14;
end;
