let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let f be PartFunc of X,REAL; :: thesis: for a being Real
for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let a be Real; :: thesis: for k being positive Real
for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let k be positive Real; :: thesis: for u being VECTOR of (RLSp_LpFunct (M,k)) st f = u holds
a (#) f = a * u
let u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( f = u implies a (#) f = a * u )
reconsider u2 = u as VECTOR of (RLSp_PFunct X) by TARSKI:def 3;
reconsider h = a * u2 as Element of PFuncs (X,REAL) ;
assume A1: f = u ; :: thesis: a (#) f = a * u
then A2: dom h = dom f by LPSPACE1:9;
then for x being object st x in dom h holds
h . x = a * (f . x) by A1, LPSPACE1:9;
then h = a (#) f by A2, VALUED_1:def 5;
hence a (#) f = a * u by LPSPACE1:5; :: thesis: verum