let X, Y, Z be RealNormSpace; :: thesis: for f being Element of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for g, h being Element of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) holds (g + h) * f = (g * f) + (h * f)
let f be Element of (R_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for g, h being Element of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) holds (g + h) * f = (g * f) + (h * f)
let g, h be Element of (R_NormSpace_of_BoundedLinearOperators (Y,Z)); :: thesis: (g + h) * f = (g * f) + (h * f)
set BLOPXY = R_NormSpace_of_BoundedLinearOperators (X,Y);
set BLOPXZ = R_NormSpace_of_BoundedLinearOperators (X,Z);
set BLOPYZ = R_NormSpace_of_BoundedLinearOperators (Y,Z);
set mf = modetrans (f,X,Y);
set mg = modetrans (g,Y,Z);
set mh = modetrans (h,Y,Z);
set mgh = modetrans ((g + h),Y,Z);
A1: (modetrans (h,Y,Z)) * (modetrans (f,X,Y)) is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
then reconsider hf = (modetrans (h,Y,Z)) * (modetrans (f,X,Y)) as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Z)) by LOPBAN_1:def 9;
A2: (modetrans (g,Y,Z)) * (modetrans (f,X,Y)) is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
then reconsider gf = (modetrans (g,Y,Z)) * (modetrans (f,X,Y)) as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Z)) by LOPBAN_1:def 9;
A3: (modetrans ((g + h),Y,Z)) * (modetrans (f,X,Y)) is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
then reconsider k = (modetrans ((g + h),Y,Z)) * (modetrans (f,X,Y)) as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Z)) by LOPBAN_1:def 9;
reconsider hh = h as VECTOR of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) ;
reconsider gg = g as VECTOR of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) ;
A4: ( gg = modetrans (g,Y,Z) & hh = modetrans (h,Y,Z) ) by LOPBAN_1:def 11;
for x being VECTOR of X holds ((modetrans ((g + h),Y,Z)) * (modetrans (f,X,Y))) . x = (((modetrans (g,Y,Z)) * (modetrans (f,X,Y))) . x) + (((modetrans (h,Y,Z)) * (modetrans (f,X,Y))) . x)

proof

let x be VECTOR of X; :: thesis: ((modetrans ((g + h),Y,Z)) * (modetrans (f,X,Y))) . x = (((modetrans (g,Y,Z)) * (modetrans (f,X,Y))) . x) + (((modetrans (h,Y,Z)) * (modetrans (f,X,Y))) . x)
( g + h = gg + hh & modetrans ((g + h),Y,Z) = g + h ) by LOPBAN_1:def 11;
then A5: (modetrans ((g + h),Y,Z)) . ((modetrans (f,X,Y)) . x) = ((modetrans (g,Y,Z)) . ((modetrans (f,X,Y)) . x)) + ((modetrans (h,Y,Z)) . ((modetrans (f,X,Y)) . x)) by A4, LOPBAN_1:35;
thus ((modetrans ((g + h),Y,Z)) * (modetrans (f,X,Y))) . x = (modetrans ((g + h),Y,Z)) . ((modetrans (f,X,Y)) . x) by A3, LPB2Th4
.= (((modetrans (g,Y,Z)) * (modetrans (f,X,Y))) . x) + ((modetrans (h,Y,Z)) . ((modetrans (f,X,Y)) . x)) by A2, A5, LPB2Th4
.= (((modetrans (g,Y,Z)) * (modetrans (f,X,Y))) . x) + (((modetrans (h,Y,Z)) * (modetrans (f,X,Y))) . x) by A1, LPB2Th4 ; :: thesis: verum

end;

hence (g + h) * f = (g * f) + (h * f) by LOPBAN_1:35; :: thesis: verum