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