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

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