let a be Real; :: thesis: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for u being VECTOR of (RLSp_AlmostZeroFunct M) st f = u holds
a (#) f = a * u
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 u being VECTOR of (RLSp_AlmostZeroFunct M) 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 u being VECTOR of (RLSp_AlmostZeroFunct M) st f = u holds
a (#) f = a * u
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for u being VECTOR of (RLSp_AlmostZeroFunct M) st f = u holds
a (#) f = a * u
let f be PartFunc of X,REAL ; :: thesis: for u being VECTOR of (RLSp_AlmostZeroFunct M) st f = u holds
a (#) f = a * u
let u be VECTOR of (RLSp_AlmostZeroFunct M); :: thesis: ( f = u implies a (#) f = a * u )
assume A1: f = u ; :: thesis: a (#) f = a * u
reconsider u2 = u as VECTOR of (RLSp_L1Funct M) by TARSKI:def 3;
thus a * u = a * u2 by RLSUB122
.= a (#) f by ThB11, A1 ; :: thesis: verum