
theorem Th14:
  for X be RealNormSpace holds Ring_of_BoundedLinearOperators(X) is Ring
proof
  let X be RealNormSpace;
  set R = Ring_of_BoundedLinearOperators X;
A1: R is right_complementable
  proof
    let x be Element of R;
    thus ex t being Element of R st x+t= 0.R by Th13;
  end;
  for x,y,z being Element of R holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.
  Ring_of_BoundedLinearOperators(X)) = x & (ex t being Element of
Ring_of_BoundedLinearOperators(X) st x+t= 0.Ring_of_BoundedLinearOperators(X))
  & (x*y)*z = x*(y*z) & x*(1.Ring_of_BoundedLinearOperators(X)) = x & (1.
Ring_of_BoundedLinearOperators(X))*x = x & x*(y+z) = x*y + x*z & (y+z)*x = y*x
  + z*x by Th13;
  hence thesis by A1,GROUP_1:def 3,RLVECT_1:def 2,def 3,def 4,VECTSP_1:def 6
,def 7;
end;
