
theorem
  for X be RealUnitarySpace holds
    DualSp X = DualSp (RUSp2RNSp X)
proof
  let X be RealUnitarySpace;
  set Y = RUSp2RNSp X;
  set X1 = BoundedLinearFunctionals X;
  set Y1 = BoundedLinearFunctionals Y;
A0: the carrier of X*' = the carrier of Y*' by LM10B;
A2: the ZeroF of DualSp X = 0.(X*') by RSSPACE:def 10
     .= 0.(Y*') by LM10B
     .= the ZeroF of DualSp Y by DUALSP01:17,RSSPACE:def 10;
A3: the addF of DualSp X
      = (the addF of X*') ||X1 by RSSPACE:def 8
     .= (the addF of Y*') ||Y1 by LM6,A0
     .= the addF of DualSp Y by DUALSP01:17,RSSPACE:def 8;
A4: the Mult of DualSp X
      = (the Mult of X*') | [:REAL,X1:] by RSSPACE:def 9
     .= (the Mult of Y*') | [:REAL,Y1:] by LM6,A0
     .= the Mult of DualSp Y by DUALSP01:17,RSSPACE:def 9;
  the normF of DualSp X = the normF of DualSp Y by LM9;
  hence DualSp X = DualSp (RUSp2RNSp X) by A2,A3,A4;
end;
