
theorem Th9:
  for X be ComplexNormSpace for f,g,h be Element of
  BoundedLinearOperators(X,X) holds f *(g+h)=f*g + f*h
proof
  let X be ComplexNormSpace;
  let f,g,h be Element of BoundedLinearOperators(X,X);
  set BLOP=C_NormSpace_of_BoundedLinearOperators(X,X);
  set ADD=Add_(BoundedLinearOperators(X,X), C_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.(mf*mg, mf*mh) =mf*mgh
  proof
    reconsider fh=mf*mh as VECTOR of BLOP by CLOPBAN1:def 7;
    reconsider fg=mf*mg as VECTOR of BLOP by CLOPBAN1:def 7;
    reconsider k=mf*mgh as VECTOR of BLOP by CLOPBAN1:def 7;
    reconsider hh = h as VECTOR of BLOP;
    reconsider gg = g as VECTOR of BLOP;
A1: gg=mg & hh=mh by CLOPBAN1:def 9;
    for x be VECTOR of X holds (mf*mgh).x=(mf*mg).x + (mf*mh).x
    proof
      let x be VECTOR of X;
      g+h=gg+hh & modetrans(g+h, X,X) =g+h by CLOPBAN1:def 9;
      then
A2:   mgh.x=mg.x+mh.x by A1,CLOPBAN1:34;
      thus (mf*mgh).x=mf.(mgh.x) by Th4
        .=mf.(mg.x) +mf.(mh.x) by A2,VECTSP_1:def 20
        .=(mf*mg).x+mf.(mh.x) by Th4
        .=(mf*mg).x+ (mf*mh).x by Th4;
    end;
    then k=fg+fh by CLOPBAN1:34;
    hence thesis;
  end;
  hence thesis;
end;
