reserve r,s,t,u for Real;

theorem Th4:
  for X being non empty addLoopStr, M,N being Subset of X, F being
Subset-Family of X st F = {x+N where x is Point of X: x in M} holds M+N = union
  F
proof
  let X be non empty addLoopStr, M,N be Subset of X, F be Subset-Family of X;
  assume
A1: F = {x+N where x is Point of X: x in M};
  thus M+N c= union F
  proof
    let x be object;
    assume x in M+N;
    then x in {u+v where u,v is Point of X: u in M & v in N} by RUSUB_4:def 9;
    then consider u,v being Point of X such that
A2: x = u+v and
A3: u in M and
A4: v in N;
    u+N = {u + v9 where v9 is Point of X: v9 in N} by RUSUB_4:def 8;
    then
A5: x in u+N by A2,A4;
    u+N in F by A1,A3;
    hence thesis by A5,TARSKI:def 4;
  end;
  let x be object;
  assume x in union F;
  then consider Y being set such that
A6: x in Y and
A7: Y in F by TARSKI:def 4;
  consider u being Point of X such that
A8: Y = u+N and
A9: u in M by A1,A7;
  u+N = {u + v where v is Point of X: v in N} by RUSUB_4:def 8;
  then ex v being Point of X st x = u+v & v in N by A6,A8;
  then x in {u9+v9 where u9,v9 is Point of X: u9 in M & v9 in N} by A9;
  hence thesis by RUSUB_4:def 9;
end;
