
theorem LM10A:
  for X be RealUnitarySpace holds
    LinearFunctionals X = LinearFunctionals (RUSp2RNSp X)
proof
  let X be RealUnitarySpace;
  set Y = RUSp2RNSp X;
  now let x be object;
    assume x in LinearFunctionals X; then
    x is linear-Functional of X by DUALSP01:def 6; then
    x is linear-Functional of Y by LM6A;
    hence x in LinearFunctionals Y by DUALSP01:def 6;
  end; then
A1: LinearFunctionals X c= LinearFunctionals Y;
  now let x be object;
    assume x in LinearFunctionals Y; then
    x is linear-Functional of Y by DUALSP01:def 6; then
    x is linear-Functional of X by LM6A;
    hence x in LinearFunctionals X by DUALSP01:def 6;
  end; then
  LinearFunctionals Y c= LinearFunctionals X;
  hence thesis by A1;
end;
