
theorem LM9:
  for X be RealUnitarySpace holds
    BoundedLinearFunctionalsNorm X
      = BoundedLinearFunctionalsNorm (RUSp2RNSp X)
proof
  let X be RealUnitarySpace;
  set Y = RUSp2RNSp X;
  set f = BoundedLinearFunctionalsNorm X;
  set g = BoundedLinearFunctionalsNorm Y;
A1: dom f = BoundedLinearFunctionals X by FUNCT_2:def 1
         .= BoundedLinearFunctionals Y by LM6
         .= dom g by FUNCT_2:def 1;
  now let x be object;
    assume B11: x in dom f; then
B1: x in BoundedLinearFunctionals X;
B2: f.x = upper_bound PreNorms(Bound2Lipschitz(x,X)) by B11,Def14;
B31: x in BoundedLinearFunctionals Y by B1,LM6;
    Bound2Lipschitz(x,X) = Bound2Lipschitz(x,RUSp2RNSp X) by LM6; then
    upper_bound PreNorms(Bound2Lipschitz(x,X))
      = upper_bound PreNorms(Bound2Lipschitz(x,Y)) by LM8;
    hence f.x = g.x by B2,B31,DUALSP01:def 14;
  end;
  hence thesis by A1;
end;
