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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (M,k)) st f = u holds

a (#) f = a * u

let k be positive Real; :: thesis: for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds

a (#) f = a * u

let u be VECTOR of (RLSp_AlmostZeroLpFunct (M,k)); :: thesis: ( f = u implies a (#) f = a * u )

reconsider u2 = u as VECTOR of (RLSp_LpFunct (M,k)) by TARSKI:def 3;

assume A1: f = u ; :: thesis: a (#) f = a * u

a * u = a * u2 by LPSPACE1:5;

hence a * u = a (#) f by Th30, A1; :: 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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (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_AlmostZeroLpFunct (M,k)) st f = u holds

a (#) f = a * u

let k be positive Real; :: thesis: for u being VECTOR of (RLSp_AlmostZeroLpFunct (M,k)) st f = u holds

a (#) f = a * u

let u be VECTOR of (RLSp_AlmostZeroLpFunct (M,k)); :: thesis: ( f = u implies a (#) f = a * u )

reconsider u2 = u as VECTOR of (RLSp_LpFunct (M,k)) by TARSKI:def 3;

assume A1: f = u ; :: thesis: a (#) f = a * u

a * u = a * u2 by LPSPACE1:5;

hence a * u = a (#) f by Th30, A1; :: thesis: verum