reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem Th46:
  r in X iff q3+r in q3++X
proof
  thus r in X implies q3+r in q3++X by MEMBER_1:141;
  assume q3+r in q3++X; then
  q3+r in { q3 + r3 : r3 in X } by Lm5;
  then ex mr being Real st q3+r = q3+mr & mr in X;
  hence thesis;
end;
