let X be ComplexNormSpace; :: thesis: for f, g, h being Element of BoundedLinearOperators (X,X) holds (g + h) * f = (g * f) + (h * f)
let f, g, h be Element of BoundedLinearOperators (X,X); :: thesis: (g + h) * f = (g * f) + (h * f)
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_ ((BoundedLinearOperators (X,X)),(C_VectorSpace_of_LinearOperators (X,X)))) . (((modetrans (g,X,X)) * (modetrans (f,X,X))),((modetrans (h,X,X)) * (modetrans (f,X,X)))) = (modetrans ((g + h),X,X)) * (modetrans (f,X,X))

proof

reconsider hf = (modetrans (h,X,X)) * (modetrans (f,X,X)) as VECTOR of (C_NormSpace_of_BoundedLinearOperators (X,X)) by CLOPBAN1:def 7;
reconsider gf = (modetrans (g,X,X)) * (modetrans (f,X,X)) as VECTOR of (C_NormSpace_of_BoundedLinearOperators (X,X)) by CLOPBAN1:def 7;
reconsider k = (modetrans ((g + h),X,X)) * (modetrans (f,X,X)) as VECTOR of (C_NormSpace_of_BoundedLinearOperators (X,X)) by CLOPBAN1:def 7;
reconsider hh = h as VECTOR of (C_NormSpace_of_BoundedLinearOperators (X,X)) ;
reconsider gg = g as VECTOR of (C_NormSpace_of_BoundedLinearOperators (X,X)) ;
A1: ( gg = modetrans (g,X,X) & hh = modetrans (h,X,X) ) by CLOPBAN1:def 9;
for x being VECTOR of X holds ((modetrans ((g + h),X,X)) * (modetrans (f,X,X))) . x = (((modetrans (g,X,X)) * (modetrans (f,X,X))) . x) + (((modetrans (h,X,X)) * (modetrans (f,X,X))) . x)

proof

let x be VECTOR of X; :: thesis: ((modetrans ((g + h),X,X)) * (modetrans (f,X,X))) . x = (((modetrans (g,X,X)) * (modetrans (f,X,X))) . x) + (((modetrans (h,X,X)) * (modetrans (f,X,X))) . x)
( g + h = gg + hh & modetrans ((g + h),X,X) = g + h ) by CLOPBAN1:def 9;
then A2: (modetrans ((g + h),X,X)) . ((modetrans (f,X,X)) . x) = ((modetrans (g,X,X)) . ((modetrans (f,X,X)) . x)) + ((modetrans (h,X,X)) . ((modetrans (f,X,X)) . x)) by A1, CLOPBAN1:34;
thus ((modetrans ((g + h),X,X)) * (modetrans (f,X,X))) . x = (modetrans ((g + h),X,X)) . ((modetrans (f,X,X)) . x) by Th4
.= (((modetrans (g,X,X)) * (modetrans (f,X,X))) . x) + ((modetrans (h,X,X)) . ((modetrans (f,X,X)) . x)) by A2, Th4
.= (((modetrans (g,X,X)) * (modetrans (f,X,X))) . x) + (((modetrans (h,X,X)) * (modetrans (f,X,X))) . x) by Th4 ; :: thesis: verum

end;

then k = gf + hf by CLOPBAN1:34;
hence (Add_ ((BoundedLinearOperators (X,X)),(C_VectorSpace_of_LinearOperators (X,X)))) . (((modetrans (g,X,X)) * (modetrans (f,X,X))),((modetrans (h,X,X)) * (modetrans (f,X,X)))) = (modetrans ((g + h),X,X)) * (modetrans (f,X,X)) ; :: thesis: verum

end;

hence (g + h) * f = (g * f) + (h * f) ; :: thesis: verum