let X, Y, Z be RealNormSpace; :: thesis: for f being Element of (R_NormSpace_of_BoundedLinearOperators (Y,Z))
for g being Element of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a, b being Real holds (a * b) * (f * g) = (a * f) * (b * g)
let f be Element of (R_NormSpace_of_BoundedLinearOperators (Y,Z)); :: thesis: for g being Element of (R_NormSpace_of_BoundedLinearOperators (X,Y))
for a, b being Real holds (a * b) * (f * g) = (a * f) * (b * g)
let g be Element of (R_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for a, b being Real holds (a * b) * (f * g) = (a * f) * (b * g)
let a, b be Real; :: thesis: (a * b) * (f * g) = (a * f) * (b * g)
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,Y,Z);
set mg = modetrans (g,X,Y);
set maf = modetrans ((a * f),Y,Z);
set mbg = modetrans ((b * g),X,Y);
A1: (modetrans ((a * f),Y,Z)) * (modetrans ((b * g),X,Y)) is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
then reconsider k = (modetrans ((a * f),Y,Z)) * (modetrans ((b * g),X,Y)) as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Z)) by LOPBAN_1:def 9;
A2: (modetrans (f,Y,Z)) * (modetrans (g,X,Y)) is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
then reconsider fg = (modetrans (f,Y,Z)) * (modetrans (g,X,Y)) as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Z)) by LOPBAN_1:def 9;
reconsider ff = f as VECTOR of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) ;
reconsider gg = g as VECTOR of (R_NormSpace_of_BoundedLinearOperators (X,Y)) ;
A3: gg = modetrans (g,X,Y) by LOPBAN_1:def 11;
A4: ff = modetrans (f,Y,Z) by LOPBAN_1:def 11;
for x being VECTOR of X holds ((modetrans ((a * f),Y,Z)) * (modetrans ((b * g),X,Y))) . x = (a * b) * (((modetrans (f,Y,Z)) * (modetrans (g,X,Y))) . x)

proof

let x be VECTOR of X; :: thesis: ((modetrans ((a * f),Y,Z)) * (modetrans ((b * g),X,Y))) . x = (a * b) * (((modetrans (f,Y,Z)) * (modetrans (g,X,Y))) . x)
set y = b * ((modetrans (g,X,Y)) . x);
( a * f = a * ff & modetrans ((a * f),Y,Z) = a * f ) by LOPBAN_1:def 11;
then A5: (modetrans ((a * f),Y,Z)) . (b * ((modetrans (g,X,Y)) . x)) = a * ((modetrans (f,Y,Z)) . (b * ((modetrans (g,X,Y)) . x))) by A4, LOPBAN_1:36;
( b * g = b * gg & modetrans ((b * g),X,Y) = b * g ) by LOPBAN_1:def 11;
then A6: (modetrans ((b * g),X,Y)) . x = b * ((modetrans (g,X,Y)) . x) by A3, LOPBAN_1:36;
thus ((modetrans ((a * f),Y,Z)) * (modetrans ((b * g),X,Y))) . x = (modetrans ((a * f),Y,Z)) . ((modetrans ((b * g),X,Y)) . x) by A1, LPB2Th4
.= a * (b * ((modetrans (f,Y,Z)) . ((modetrans (g,X,Y)) . x))) by A5, A6, LOPBAN_1:def 5
.= (a * b) * ((modetrans (f,Y,Z)) . ((modetrans (g,X,Y)) . x)) by RLVECT_1:def 7
.= (a * b) * (((modetrans (f,Y,Z)) * (modetrans (g,X,Y))) . x) by A2, LPB2Th4 ; :: thesis: verum

end;

hence (a * b) * (f * g) = (a * f) * (b * g) by LOPBAN_1:36; :: thesis: verum