
theorem Th20:
  for X be RealNormSpace holds R_Normed_Algebra_of_BoundedLinearOperators(X) is
  reflexive discerning RealNormSpace-like Abelian
  add-associative right_zeroed right_complementable associative right_unital
  right-distributive vector-distributive scalar-distributive scalar-associative
  vector-associative vector-distributive scalar-distributive scalar-associative
  scalar-unital
proof
  let X be RealNormSpace;
  set RBLOP=R_Normed_Algebra_of_BoundedLinearOperators(X);
  set BS=R_NormSpace_of_BoundedLinearOperators(X,X);
  set ADD=Add_(BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X
  ,X));
  set EXMULT =Mult_(BoundedLinearOperators(X,X),
  R_VectorSpace_of_LinearOperators(X,X));
  set NRM=BoundedLinearOperatorsNorm(X,X);
A1: ||.0.BS.|| = NRM.(0.BS)
      .= ||.0.RBLOP.||;
  thus ||.0.RBLOP.|| = 0 by A1;
A2: now
    let x,y be Point of RBLOP;
    let a be Real;
    reconsider x1 =x, y1 =y as Point of BS;
A3: ||.x1.|| + ||. y1.|| = NRM.(x1) + ||. y1.||
      .= NRM.(x1) + NRM.(y1)
      .= ||.x.|| +( the normF of RBLOP ).y
      .= ||.x.|| + ||.y.||;
    ||.x + y.|| = NRM.(ADD.(x,y))
      .= ||.x1 + y1.||;
    hence ||.x + y.|| <= ||.x.|| + ||. y.|| by A3,NORMSP_1:def 1;
A4: ||.x1.|| = NRM.(x1)
      .= ||.x.||;
    0.BS=0.RBLOP;
    hence ||.x.|| = 0 iff x= 0.RBLOP by A4,NORMSP_0:def 5;
    thus ||.a*x.|| = NRM.(EXMULT.(a,x))
      .= ||.a*x1.||
      .=|.a.|* ||.x.|| by A4,NORMSP_1:def 1;
  end;
A5: RBLOP is right_complementable
  proof
    let x be Element of RBLOP;
    thus ex t being Element of RBLOP st x+t= 0.RBLOP by Th19;
  end;
  ( for x,y,z being Element of R_Normed_Algebra_of_BoundedLinearOperators(
  X) for a,b be Real holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.
  R_Normed_Algebra_of_BoundedLinearOperators(X)) = x & (ex t being Element of
  R_Normed_Algebra_of_BoundedLinearOperators(X) st x+t= 0.
  R_Normed_Algebra_of_BoundedLinearOperators(X)) & (x*y)*z = x*(y*z) & x*(1.
  R_Normed_Algebra_of_BoundedLinearOperators(X)) = x & (1.
R_Normed_Algebra_of_BoundedLinearOperators(X))*x = x & x*(y+z) = x*y + x*z & (y
+z)*x = y*x + z*x & a*(x*y) = (a*x)*y & (a*b)*(x*y)=(a*x)*(b*y) & a*(x+y) = a*x
+ a*y & (a+b)*x = a*x + b*x & (a*b)*x = a*(b*x) & 1*x=x)& for a be Real
  for v,w being VECTOR of RBLOP holds a * (v + w) = a * v + a * w by Th19;
  hence thesis by A5,A2,NORMSP_1:def 1;
end;
