
theorem Th10:
  for X be RealNormSpace for f,g,h be Element of
  BoundedLinearOperators(X,X) holds (g+h)*f = g*f + h*f
proof
  let X be RealNormSpace;
  let f,g,h be Element of BoundedLinearOperators(X,X);
  set BLOP=R_NormSpace_of_BoundedLinearOperators(X,X);
  set ADD=Add_(BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X
  ,X));
  set mf=modetrans(f,X,X);
  set mg=modetrans(g,X,X);
  set mh=modetrans(h,X,X);
  set mgh=modetrans(g+h, X,X);
  ADD.(mg*mf, mh*mf) =mgh*mf
  proof
    reconsider hf=mh*mf as VECTOR of BLOP by LOPBAN_1:def 9;
    reconsider gf=mg*mf as VECTOR of BLOP by LOPBAN_1:def 9;
    reconsider k=mgh*mf as VECTOR of BLOP by LOPBAN_1:def 9;
    reconsider hh = h as VECTOR of BLOP;
    reconsider gg = g as VECTOR of BLOP;
A1: gg=mg & hh=mh by LOPBAN_1:def 11;
    for x be VECTOR of X holds (mgh*mf).x=(mg*mf).x + (mh*mf).x
    proof
      let x be VECTOR of X;
      g+h=gg+hh & modetrans(g+h, X,X) =g+h by LOPBAN_1:def 11;
      then
A2:   mgh.(mf.x)=mg.(mf.x)+mh.(mf.x) by A1,LOPBAN_1:35;
      thus (mgh*mf).x=mgh.(mf.x) by Th4
        .=(mg*mf).x+mh.(mf.x) by A2,Th4
        .=(mg*mf).x+ (mh*mf).x by Th4;
    end;
    then k=gf+hf by LOPBAN_1:35;
    hence thesis;
  end;
  hence thesis;
end;
