let a be Complex; :: 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,COMPLEX
for u being VECTOR of (CLSp_L1Funct 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,COMPLEX
for u being VECTOR of (CLSp_L1Funct 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,COMPLEX
for u being VECTOR of (CLSp_L1Funct M) st f = u holds
a (#) f = a * u
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,COMPLEX
for u being VECTOR of (CLSp_L1Funct M) st f = u holds
a (#) f = a * u
let f be PartFunc of X,COMPLEX; :: thesis: for u being VECTOR of (CLSp_L1Funct M) st f = u holds
a (#) f = a * u
let u be VECTOR of (CLSp_L1Funct M); :: thesis: ( f = u implies a (#) f = a * u )
reconsider u2 = u as VECTOR of (CLSp_PFunct X) by TARSKI:def 3;
reconsider h = a * u2 as Element of PFuncs (X,COMPLEX) ;
assume A1: f = u ; :: thesis: a (#) f = a * u
A2: a * u2 = (multcomplexcpfunc X) . (a,u2) ;
A3: dom h = dom f by A1, A2, Th7;
A4: for x being object st x in dom h holds
h . x = a * (f . x) by A1, A2, A3, Th7;
A5: h = a (#) f by A3, A4, VALUED_1:def 5;
reconsider a = a as Element of COMPLEX by XCMPLX_0:def 2;
[a,u] in [:COMPLEX,(L1_CFunctions M):] ;
hence a (#) f = a * u by A5, FUNCT_1:49; :: thesis: verum